title: Group (mathematics)

text: together with addition. For any two integers formula_2 and formula_3, the sum formula_4 is also an integer; this "closure" property says that formula_5 is a binary operation on formula_6. The following properties of integer addition serve as a model for the group axioms in the definition below.


title: Group (mathematics)

text: The integers, together with the operation formula_5, form a mathematical object belonging to a broad class sharing similar structural aspects. To appropriately understand these structures as a collective, the following definition is developed.


title: Group (mathematics)

text: A group is a set formula_28 together with a binary operation on formula_28, here denoted "formula_30", that combines any two elements formula_2 and formula_3 to form an element of formula_28, denoted formula_34, such that the following three requirements, known as "group axioms", are satisfied:


title: Group (mathematics)

text: Formally, the group is the ordered pair of a set and a binary operation on this set that satisfies the group axioms. The set is called the "underlying set" of the group, and the operation is called the "group operation" or the "group law".


title: Group (mathematics)

text: A group and its underlying set are thus two different mathematical objects. To avoid cumbersome notation, it is common to abuse notation by using the same symbol to denote both. This reflects also an informal way of thinking: that the group is the same as the set except that it has been enriched by additional structure provided by the operation.


title: Group (mathematics)

text: For example, consider the set of real numbers formula_57, which has the operations of addition formula_4 and multiplication formula_59. Formally, formula_57 is a set, formula_61 is a group, and formula_62 is a field. But it is common to write formula_57 to denote any of these three objects.


title: Group (mathematics)

text: The "additive group" of the field formula_57 is the group whose underlying set is formula_57 and whose operation is addition. The "multiplicative group" of the field formula_57 is the group formula_67 whose underlying set is the set of nonzero real numbers formula_68 and whose operation is multiplication.


title: Group (mathematics)

text: More generally, one speaks of an "additive group" whenever the group operation is notated as addition; in this case, the identity is typically denoted formula_69, and the inverse of an element formula_70 is denoted formula_71. Similarly, one speaks of a "multiplicative group" whenever the group operation is notated as multiplication; in this case, the identity is typically denoted formula_72, and the inverse of an element formula_70 is denoted formula_74. In a multiplicative group, the operation symbol is usually omitted entirely, so that the operation is denoted by juxtaposition, formula_59 instead of formula_34.


title: Group (mathematics)

text: The definition of a group does not require that formula_77 for all elements formula_2 and formula_3 in formula_28. If this additional condition holds, then the operation is said to be commutative, and the group is called an abelian group. It is a common convention that for an abelian group either additive or multiplicative notation may be used, but for a nonabelian group only multiplicative notation is used.


title: Group (mathematics)

text: Several other notations are commonly used for groups whose elements are not numbers. For a group whose elements are functions, the operation is often function composition formula_81; then the identity may be denoted id. In the more specific cases of geometric transformation groups, symmetry groups, permutation groups, and automorphism groups, the symbol formula_82 is often omitted, as for multiplicative groups. Many other variants of notation may be encountered.


title: Group (mathematics)

text: Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries. A square has eight symmetries. These are:


title: Group (mathematics)

text: These symmetries are functions. Each sends a point in the square to the corresponding point under the symmetry. For example, formula_83 sends a point to its rotation 90° clockwise around the square's center, and formula_87 sends a point to its reflection across the square's vertical middle line. Composing two of these symmetries gives another symmetry. These symmetries determine a group called the dihedral group of degree four, denoted formula_92. The underlying set of the group is the above set of symmetries, and the group operation is function composition. Two symmetries are combined by composing them as functions, that is, applying the first one to the square, and the second one to the result of the first application. The result of performing first formula_2 and then formula_3 is written symbolically "from right to left" as formula_95 ("apply the symmetry formula_3 after performing the symmetry formula_2"). This is the usual notation for composition of functions.


title: Group (mathematics)

text: The group table lists the results of all such compositions possible. For example, rotating by 270° clockwise (formula_85) and then reflecting horizontally (formula_87) is the same as performing a reflection along the diagonal (formula_88). Using the above symbols, highlighted in blue in the group table:


title: Group (mathematics)

text: Given this set of symmetries and the described operation, the group axioms can be understood as follows.


title: Group (mathematics)

text: "Binary operation": Composition is a binary operation. That is, formula_102 is a symmetry for any two symmetries formula_2 and formula_3. For example,


title: Group (mathematics)

text: that is, rotating 270° clockwise after reflecting horizontally equals reflecting along the counter-diagonal (formula_89). Indeed, every other combination of two symmetries still gives a symmetry, as can be checked using the group table.


title: Group (mathematics)

text: "Associativity": The associativity axiom deals with composing more than two symmetries: Starting with three elements formula_2, formula_3 and formula_9 of formula_92, there are two possible ways of using these three symmetries in this order to determine a symmetry of the square. One of these ways is to first compose formula_2 and formula_3 into a single symmetry, then to compose that symmetry with formula_9. The other way is to first compose formula_3 and formula_9, then to compose the resulting symmetry with formula_2. These two ways must give always the same result, that is,


title: Group (mathematics)

text: "Identity element": The identity element is formula_120, as it does not change any symmetry formula_2 when composed with it either on the left or on the right.


title: Group (mathematics)

text: "Inverse element": Each symmetry has an inverse: formula_120, the reflections formula_87, formula_86, formula_88, formula_89 and the 180° rotation formula_84 are their own inverse, because performing them twice brings the square back to its original orientation. The rotations formula_85 and formula_83 are each other's inverses, because rotating 90° and then rotation 270° (or vice versa) yields a rotation over 360° which leaves the square unchanged. This is easily verified on the table.


title: Group (mathematics)

text: In contrast to the group of integers above, where the order of the operation is immaterial, it does matter in formula_92, as, for example, formula_131 but formula_132. In other words, formula_92 is not abelian.


title: Group (mathematics)

text: The modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4. The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots (solutions). The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's "On the theory of groups, as depending on the symbolic equation formula_134" (1854) gives the first abstract definition of a finite group.


title: Group (mathematics)

text: Geometry was a second field in which groups were used systematically, especially symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884.


title: Group (mathematics)

text: The third field contributing to group theory was number theory. Certain abelian group structures had been used implicitly in Carl Friedrich Gauss's number-theoretical work "Disquisitiones Arithmeticae" (1798), and more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers.


title: Group (mathematics)

text: The convergence of these various sources into a uniform theory of groups started with Camille Jordan's (1870). Walther von Dyck (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time. As of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, and more generally locally compact groups was studied by Hermann Weyl, Élie Cartan and many others. Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley (from the late 1930s) and later by the work of Armand Borel and Jacques Tits.


title: Group (mathematics)

text: The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research concerning this classification proof is ongoing. Group theory remains a highly active mathematical branch, impacting many other fields, as the examples below illustrate.


title: Group (mathematics)

text: Basic facts about all groups that can be obtained directly from the group axioms are commonly subsumed under "elementary group theory". For example, repeated applications of the associativity axiom show that the unambiguity of


title: Group (mathematics)

text: generalizes to more than three factors. Because this implies that parentheses can be inserted anywhere within such a series of terms, parentheses are usually omitted.


title: Group (mathematics)

text: Individual axioms may be "weakened" to assert only the existence of a left identity and left inverses. From these "one-sided axioms", one can prove that the left identity is also a right identity and a left inverse is also a right inverse for the same element. Since they define exactly the same structures as groups, collectively the axioms are no weaker.


title: Group (mathematics)

text: The group axioms imply that the identity element is unique: If formula_40 and formula_137 are identity elements of a group, then formula_138. Therefore, it is customary to speak of "the" identity.


title: Group (mathematics)

text: The group axioms also imply that the inverse of each element is unique: If a group element formula_2 has both formula_3 and formula_9 as inverses, then


title: Group (mathematics)

text: Given elements formula_2 and formula_3 of a group formula_28, there is a unique solution formula_70 in formula_28 to the equation formula_147, namely formula_148. (One usually avoids using fraction notation formula_149 unless formula_28 is abelian, because of the ambiguity of whether it means formula_148 or formula_152.) It follows that for each formula_2 in formula_28, the function formula_155 that maps each formula_70 to formula_157 is a bijection; it is called "left multiplication by formula_2" or "left translation by formula_2".


title: Group (mathematics)

text: Similarly, given formula_2 and formula_3, the unique solution to formula_162 is formula_152. For each formula_2, the function formula_155 that maps each formula_70 to formula_167 is a bijection called "right multiplication by formula_2" or "right translation by formula_2".


title: Group (mathematics)

text: When studying sets, one uses concepts such as subset, function, and quotient by an equivalence relation. When studying groups, one uses instead subgroups, homomorphisms, and quotient groups. These are the analogues that take the group structure into account.


title: Group (mathematics)

text: Group homomorphisms are functions that respect group structure; they may be used to relate two groups. A "homomorphism" from a group formula_170 to a group formula_171 is a function formula_172 such that


title: Group (mathematics)

text: It would be natural to require also that formula_173 respect identities, formula_174, and inverses, formula_175 for all formula_2 in formula_28. However, these additional requirements need not be included in the definition of homomorphisms, because they are already implied by the requirement of respecting the group operation.


title: Group (mathematics)

text: The "identity homomorphism" of a group formula_28 is the homomorphism formula_179 that maps each element of formula_28 to itself. An "inverse homomorphism" of a homomorphism formula_172 is a homomorphism formula_182 such that formula_183 and formula_184, that is, such that formula_185 for all formula_186 in formula_28 and such that formula_188 for all formula_189 in formula_190. An "isomorphism" is a homomorphism that has an inverse homomorphism; equivalently, it is a bijective homomorphism. Groups formula_28 and formula_190 are called "isomorphic" if there exists an isomorphism formula_172. In this case, formula_190 can be obtained from formula_28 simply by renaming its elements according to the function formula_173; then any statement true for formula_28 is true for formula_190, provided that any specific elements mentioned in the statement are also renamed.


title: Group (mathematics)

text: The collection of all groups, together with the homomorphisms between them, form a category, the category of groups.


title: Group (mathematics)

text: Informally, a "subgroup" is a group formula_190 contained within a bigger one, formula_28: it has a subset of the elements of formula_28, with the same operation. Concretely, this means that the identity element of formula_28 must be contained in formula_190, and whenever formula_204 and formula_205 are both in formula_190, then so are formula_207 and formula_208, so the elements of formula_190, equipped with the group operation on formula_28 restricted to formula_190, indeed form a group. In this case, the inclusion map formula_212 is a homomorphism.


title: Group (mathematics)

text: In the example of symmetries of a square, the identity and the rotations constitute a subgroup formula_213, highlighted in red in the group table of the example: any two rotations composed are still a rotation, and a rotation can be undone by (i.e., is inverse to) the complementary rotations 270° for 90°, 180° for 180°, and 90° for 270°. The subgroup test provides a necessary and sufficient condition for a nonempty subset "H" of a group "G" to be a subgroup: it is sufficient to check that formula_214 for all elements formula_186 and formula_189 in formula_190. Knowing a group's subgroups is important in understanding the group as a whole.


title: Group (mathematics)

text: Given any subset formula_218 of a group formula_28, the subgroup generated by formula_218 consists of all products of elements of formula_218 and their inverses. It is the smallest subgroup of formula_28 containing formula_218. In the example of symmetries of a square, the subgroup generated by formula_84 and formula_86 consists of these two elements, the identity element formula_120, and the element formula_227. Again, this is a subgroup, because combining any two of these four elements or their inverses (which are, in this particular case, these same elements) yields an element of this subgroup.


title: Group (mathematics)

text: An injective homomorphism formula_228 factors canonically as an isomorphism followed by an inclusion, formula_229 for some subgroup of .


title: Group (mathematics)

text: In many situations it is desirable to consider two group elements the same if they differ by an element of a given subgroup. For example, in the symmetry group of a square, once any reflection is performed, rotations alone cannot return the square to its original position, so one can think of the reflected positions of the square as all being equivalent to each other, and as inequivalent to the unreflected positions; the rotation operations are irrelevant to the question whether a reflection has been performed. Cosets are used to formalize this insight: a subgroup formula_190 determines left and right cosets, which can be thought of as translations of formula_190 by an arbitrary group element formula_186. In symbolic terms, the "left" and "right" cosets of formula_190, containing an element formula_186, are


title: Group (mathematics)

text: The left cosets of any subgroup formula_190 form a partition of formula_28; that is, the union of all left cosets is equal to formula_28 and two left cosets are either equal or have an empty intersection. The first case formula_238 happens precisely when formula_239, i.e., when the two elements differ by an element of formula_190. Similar considerations apply to the right cosets of formula_190. The left cosets of formula_190 may or may not be the same as its right cosets. If they are (that is, if all formula_186 in formula_28 satisfy formula_245), then formula_190 is said to be a "normal subgroup".


title: Group (mathematics)

text: In formula_92, the group of symmetries of a square, with its subgroup formula_248 of rotations, the left cosets formula_249 are either equal to formula_248, if formula_186 is an element of formula_248 itself, or otherwise equal to formula_253 (highlighted in green in the group table of formula_92). The subgroup formula_248 is normal, because formula_256 and similarly for the other elements of the group. (In fact, in the case of formula_92, the cosets generated by reflections are all equal: formula_258.)


title: Group (mathematics)

text: Then there is a unique group law on formula_262 for which the map formula_263 sending each element formula_186 to formula_265 is a homomorphism.


title: Group (mathematics)

text: Explicitly, the product of two cosets formula_265 and formula_267 is formula_268, the coset formula_269 serves as the identity of formula_262, and the inverse of formula_265 in the quotient group is .


title: Group (mathematics)

text: The group formula_262, read as "formula_28 modulo formula_259", is called a "quotient group" or "factor group".


title: Group (mathematics)

text: The elements of the quotient group formula_275 are formula_248 and formula_277. The group operation on the quotient is shown in the table. For example, formula_278. Both the subgroup formula_213 and the quotient formula_275 are abelian, but formula_92 is not. Sometimes a group can be reconstructed from a subgroup and quotient (plus some additional data), by the semidirect product construction; formula_92 is an example.


title: Group (mathematics)

text: The first isomorphism theorem implies that any surjective homomorphism formula_283 factors canonically as a quotient homomorphism followed by an isomorphism: formula_284.


title: Group (mathematics)

text: For example, the dihedral group formula_92 is generated by the right rotation formula_83 and the reflection formula_86 in a vertical line (every element of formula_92 is a finite product of copies of these and their inverses).


title: Group (mathematics)

text: Hence there is a surjective homomorphism from the free group formula_289 on two generators to formula_92 sending formula_291 to formula_83 and formula_137 to formula_294.


title: Group (mathematics)

text: In fact, it turns out that formula_295 is the smallest normal subgroup of formula_289 containing these three elements; in other words, all relations are consequences of these three.


title: Group (mathematics)

text: This is called a "presentation" of formula_92 by generators and relations, because the first isomorphism theorem for yields an isomorphism formula_301.


title: Group (mathematics)

text: A presentation of a group can be used to construct the Cayley graph, a graphical depiction of a discrete group.


title: Group (mathematics)

text: Examples and applications of groups abound. A starting point is the group formula_6 of integers with addition as group operation, introduced above. If instead of addition multiplication is considered, one obtains multiplicative groups. These groups are predecessors of important constructions in abstract algebra.


title: Group (mathematics)

text: Groups are also applied in many other mathematical areas. Mathematical objects are often examined by associating groups to them and studying the properties of the corresponding groups. For example, Henri Poincaré founded what is now called algebraic topology by introducing the fundamental group. By means of this connection, topological properties such as proximity and continuity translate into properties of groups. For example, elements of the fundamental group are represented by loops. The second image shows some loops in a plane minus a point. The blue loop is considered null-homotopic (and thus irrelevant), because it can be continuously shrunk to a point. The presence of the hole prevents the orange loop from being shrunk to a point. The fundamental group of the plane with a point deleted turns out to be infinite cyclic, generated by the orange loop (or any other loop winding once around the hole). This way, the fundamental group detects the hole.


title: Group (mathematics)

text: In more recent applications, the influence has also been reversed to motivate geometric constructions by a group-theoretical background. In a similar vein, geometric group theory employs geometric concepts, for example in the study of hyperbolic groups. Further branches crucially applying groups include algebraic geometry and number theory.


title: Group (mathematics)

text: In addition to the above theoretical applications, many practical applications of groups exist. Cryptography relies on the combination of the abstract group theory approach together with algorithmical knowledge obtained in computational group theory, in particular when implemented for finite groups. Applications of group theory are not restricted to mathematics; sciences such as physics, chemistry and computer science benefit from the concept.


title: Group (mathematics)

text: Many number systems, such as the integers and the rationals, enjoy a naturally given group structure. In some cases, such as with the rationals, both addition and multiplication operations give rise to group structures. Such number systems are predecessors to more general algebraic structures known as rings and fields. Further abstract algebraic concepts such as modules, vector spaces and algebras also form groups.


title: Group (mathematics)

text: The group of integers formula_6 under addition, denoted formula_304, has been described above. The integers, with the operation of multiplication instead of addition, formula_305 do "not" form a group. The associativity and identity axioms are satisfied, but inverses do not exist: for example, formula_306 is an integer, but the only solution to the equation formula_307 in this case is formula_308, which is a rational number, but not an integer. Hence not every element of formula_6 has a (multiplicative) inverse.


title: Group (mathematics)

text: Fractions of integers (with formula_3 nonzero) are known as rational numbers. The set of all such irreducible fractions is commonly denoted formula_312. There is still a minor obstacle for formula_313, the rationals with multiplication, being a group: because zero does not have a multiplicative inverse (i.e., there is no formula_70 such that formula_315), formula_313 is still not a group.


title: Group (mathematics)

text: However, the set of all "nonzero" rational numbers formula_317 does form an abelian group under multiplication, also denoted Associativity and identity element axioms follow from the properties of integers. The closure requirement still holds true after removing zero, because the product of two nonzero rationals is never zero. Finally, the inverse of formula_318 is formula_319, therefore the axiom of the inverse element is satisfied.


title: Group (mathematics)

text: The rational numbers (including zero) also form a group under addition. Intertwining addition and multiplication operations yields more complicated structures called rings and – if division by other than zero is possible, such as in formula_312 – fields, which occupy a central position in abstract algebra. Group theoretic arguments therefore underlie parts of the theory of those entities.


title: Group (mathematics)

text: Modular arithmetic for a "modulus" formula_321 defines any two elements formula_2 and formula_3 that differ by a multiple of formula_321 to be equivalent, denoted by formula_325. Every integer is equivalent to one of the integers from formula_69 to formula_327, and the operations of modular arithmetic modify normal arithmetic by replacing the result of any operation by its equivalent representative. Modular addition, defined in this way for the integers from formula_69 to formula_327, forms a group, denoted as formula_330 or formula_331, with formula_69 as the identity element and formula_333 as the inverse element of formula_2.


title: Group (mathematics)

text: A familiar example is addition of hours on the face of a clock, where 12 rather than 0 is chosen as the representative of the identity. If the hour hand is on formula_335 and is advanced formula_336 hours, it ends up on formula_72, as shown in the illustration. This is expressed by saying that formula_338 is congruent to formula_72 "modulo formula_340" or, in symbols,


title: Group (mathematics)

text: For any prime number formula_342, there is also the multiplicative group of integers modulo formula_342. Its elements can be represented by formula_72 to formula_345. The group operation, multiplication modulo formula_342, replaces the usual product by its representative, the remainder of division by formula_342. For example, for formula_348, the four group elements can be represented by formula_349. In this group, formula_350, because the usual product formula_351 is equivalent to formula_72: when divided by formula_353 it yields a remainder of formula_72. The primality of formula_342 ensures that the usual product of two representatives is not divisible by formula_342, and therefore that the modular product is nonzero. The identity element is represented and associativity follows from the corresponding property of the integers. Finally, the inverse element axiom requires that given an integer formula_2 not divisible by formula_342, there exists an integer formula_3 such that


title: Group (mathematics)

text: that is, such that formula_342 evenly divides formula_362. The inverse formula_3 can be found by using Bézout's identity and the fact that the greatest common divisor formula_364 In the case formula_348 above, the inverse of the element represented by formula_336 is that represented by formula_336, and the inverse of the element represented by formula_368 is represented , as formula_369. Hence all group axioms are fulfilled. This example is similar to formula_370 above: it consists of exactly those elements in the ring formula_371 that have a multiplicative inverse. These groups, denoted formula_372, are crucial to public-key cryptography.


title: Group (mathematics)

text: A "cyclic group" is a group all of whose elements are powers of a particular element formula_2. In multiplicative notation, the elements of the group are


title: Group (mathematics)

text: where formula_375 means formula_376, formula_377 stands for formula_378, etc. Such an element formula_2 is called a generator or a primitive element of the group. In additive notation, the requirement for an element to be primitive is that each element of the group can be written as


title: Group (mathematics)

text: In the groups formula_331 introduced above, the element formula_72 is primitive, so these groups are cyclic. Indeed, each element is expressible as a sum all of whose terms are formula_72. Any cyclic group with formula_321 elements is isomorphic to this group. A second example for cyclic groups is the group of formula_321th complex roots of unity, given by complex numbers formula_386 satisfying formula_387. These numbers can be visualized as the vertices on a regular formula_321-gon, as shown in blue in the image for formula_389. The group operation is multiplication of complex numbers. In the picture, multiplying with formula_386 corresponds to a counter-clockwise rotation by 60°. From field theory, the group formula_372 is cyclic for prime formula_342: for example, if formula_348, formula_368 is a generator since formula_395, formula_396, formula_397, and formula_398.


title: Group (mathematics)

text: Some cyclic groups have an infinite number of elements. In these groups, for every non-zero element formula_2, all the powers of formula_2 are distinct; despite the name "cyclic group", the powers of the elements do not cycle. An infinite cyclic group is isomorphic to formula_401, the group of integers under addition introduced above. As these two prototypes are both abelian, so are all cyclic groups.


title: Group (mathematics)

text: The study of finitely generated abelian groups is quite mature, including the fundamental theorem of finitely generated abelian groups; and reflecting this state of affairs, many group-related notions, such as center and commutator, describe the extent to which a given group is not abelian.


title: Group (mathematics)

text: "Symmetry groups" are groups consisting of symmetries of given mathematical objects, principally geometric entities, such as the symmetry group of the square given as an introductory example above, although they also arise in algebra such as the symmetries among the roots of polynomial equations dealt with in Galois theory (see below). Conceptually, group theory can be thought of as the study of symmetry. Symmetries in mathematics greatly simplify the study of geometrical or analytical objects. A group is said to act on another mathematical object "X" if every group element can be associated to some operation on "X" and the composition of these operations follows the group law. For example, an element of the (2,3,7) triangle group acts on a triangular tiling of the hyperbolic plane by permuting the triangles. By a group action, the group pattern is connected to the structure of the object being acted on.


title: Group (mathematics)

text: In chemical fields, such as crystallography, space groups and point groups describe molecular symmetries and crystal symmetries. These symmetries underlie the chemical and physical behavior of these systems, and group theory enables simplification of quantum mechanical analysis of these properties. For example, group theory is used to show that optical transitions between certain quantum levels cannot occur simply because of the symmetry of the states involved.


title: Group (mathematics)

text: Group theory helps predict the changes in physical properties that occur when a material undergoes a phase transition, for example, from a cubic to a tetrahedral crystalline form. An example is ferroelectric materials, where the change from a paraelectric to a ferroelectric state occurs at the Curie temperature and is related to a change from the high-symmetry paraelectric state to the lower symmetry ferroelectric state, accompanied by a so-called soft phonon mode, a vibrational lattice mode that goes to zero frequency at the transition.


title: Group (mathematics)

text: Such spontaneous symmetry breaking has found further application in elementary particle physics, where its occurrence is related to the appearance of Goldstone bosons.


title: Group (mathematics)

text: Finite symmetry groups such as the Mathieu groups are used in coding theory, which is in turn applied in error correction of transmitted data, and in CD players. Another application is differential Galois theory, which characterizes functions having antiderivatives of a prescribed form, giving group-theoretic criteria for when solutions of certain differential equations are well-behaved. Geometric properties that remain stable under group actions are investigated in (geometric) invariant theory.


title: Group (mathematics)

text: Matrix groups consist of matrices together with matrix multiplication. The "general linear group" formula_402 consists of all invertible formula_321-by-formula_321 matrices with real entries. Its subgroups are referred to as "matrix groups" or "linear groups". The dihedral group example mentioned above can be viewed as a (very small) matrix group. Another important matrix group is the special orthogonal group formula_405. It describes all possible rotations in formula_321 dimensions. Rotation matrices in this group are used in computer graphics.


title: Group (mathematics)

text: "Representation theory" is both an application of the group concept and important for a deeper understanding of groups. It studies the group by its group actions on other spaces. A broad class of group representations are linear representations in which the group acts on a vector space, such as the three-dimensional Euclidean space formula_407. A representation of a group formula_28 on an formula_321-dimensional real vector space is simply a group homomorphism


title: Group (mathematics)

text: from the group to the general linear group. This way, the group operation, which may be abstractly given, translates to the multiplication of matrices making it accessible to explicit computations.


title: Group (mathematics)

text: A group action gives further means to study the object being acted on. On the other hand, it also yields information about the group. Group representations are an organizing principle in the theory of finite groups, Lie groups, algebraic groups and topological groups, especially (locally) compact groups.


title: Group (mathematics)

text: "Galois groups" were developed to help solve polynomial equations by capturing their symmetry features. For example, the solutions of the quadratic equation formula_411 are given by


title: Group (mathematics)

text: Each solution can be obtained by replacing the formula_413 sign by formula_5 or formula_415; analogous formulae are known for cubic and quartic equations, but do "not" exist in general for degree 5 and higher. In the quadratic formula, changing the sign (permuting the resulting two solutions) can be viewed as a (very simple) group operation. Analogous Galois groups act on the solutions of higher-degree polynomials and are closely related to the existence of formulas for their solution. Abstract properties of these groups (in particular their solvability) give a criterion for the ability to express the solutions of these polynomials using solely addition, multiplication, and roots similar to the formula above.


title: Group (mathematics)

text: Modern Galois theory generalizes the above type of Galois groups by shifting to field theory and considering field extensions formed as the splitting field of a polynomial. This theory establishes—via the fundamental theorem of Galois theory—a precise relationship between fields and groups, underlining once again the ubiquity of groups in mathematics.


title: Group (mathematics)

text: A group is called "finite" if it has a finite number of elements. The number of elements is called the order of the group. An important class is the "symmetric groups" formula_416, the groups of permutations of formula_259 objects. For example, the symmetric group on 3 letters formula_418 is the group of all possible reorderings of the objects. The three letters ABC can be reordered into ABC, ACB, BAC, BCA, CAB, CBA, forming in total 6 (factorial of 3) elements. The group operation is composition of these reorderings, and the identity element is the reordering operation that leaves the order unchanged. This class is fundamental insofar as any finite group can be expressed as a subgroup of a symmetric group formula_416 for a suitable integer formula_259, according to Cayley's theorem. Parallel to the group of symmetries of the square above, formula_418 can also be interpreted as the group of symmetries of an equilateral triangle.


title: Group (mathematics)

text: The order of an element formula_2 in a group formula_28 is the least positive integer formula_321 such that formula_425, where formula_426 represents


title: Group (mathematics)

text: that is, application of the operation "formula_30" to formula_321 copies of formula_2. (If "formula_30" represents multiplication, then formula_426 corresponds to the formula_321th power of formula_2.) In infinite groups, such an formula_321 may not exist, in which case the order of formula_2 is said to be infinity. The order of an element equals the order of the cyclic subgroup generated by this element.


title: Group (mathematics)

text: More sophisticated counting techniques, for example, counting cosets, yield more precise statements about finite groups: Lagrange's Theorem states that for a finite group formula_28 the order of any finite subgroup formula_190 divides the order of formula_28. The Sylow theorems give a partial converse.


title: Group (mathematics)

text: The dihedral group formula_92 of symmetries of a square is a finite group of order 8. In this group, the order of formula_83 is 4, as is the order of the subgroup formula_248 that this element generates. The order of the reflection elements formula_86 etc. is 2. Both orders divide 8, as predicted by Lagrange's theorem. The groups formula_372 of multiplication modulo a prime formula_342 have order formula_345.


title: Group (mathematics)

text: Any finite abelian group is isomorphic to a product of finite cyclic groups; this statement is part of the fundamental theorem of finitely generated abelian groups.


title: Group (mathematics)

text: Any group of prime order formula_342 is isomorphic to the cyclic group formula_448 (a consequence of Lagrange's theorem).


title: Group (mathematics)

text: But there exist nonabelian groups of order formula_452; the dihedral group formula_92 of order formula_454 above is an example.


title: Group (mathematics)

text: When a group formula_28 has a normal subgroup formula_259 other than formula_457 and formula_28 itself, questions about formula_28 can sometimes be reduced to questions about formula_259 and formula_262. A nontrivial group is called "simple" if it has no such normal subgroup. Finite simple groups are to finite groups as prime numbers are to positive integers: they serve as building blocks, in a sense made precise by the Jordan–Hölder theorem.


title: Group (mathematics)

text: The classification of all finite "simple" groups was a major achievement in contemporary group theory. There are several infinite families of such groups, as well as 26 "sporadic groups" that do not belong to any of the families. The largest sporadic group is called the monster group. The monstrous moonshine conjectures, proved by Richard Borcherds, relate the monster group to certain modular functions.


title: Group (mathematics)

text: The gap between the classification of simple groups and the classification of all groups lies in the extension problem.


title: Group (mathematics)

text: An equivalent definition of group consists of replacing the "there exist" part of the group axioms by operations whose result is the element that must exist. So, a group is a set formula_28 equipped with a binary operation formula_463 (the group operation), a unary operation formula_464 (which provides the inverse) and a nullary operation, which has no operand and results in the identity element. Otherwise, the group axioms are exactly the same. This variant of the definition avoids existential quantifiers and is used in computing with groups and for computer-aided proofs.


title: Group (mathematics)

text: This way of defining groups lends itself to generalizations such as the notion of group object in a category. Briefly, this is an object with morphisms that mimic the group axioms.


title: Group (mathematics)

text: Some topological spaces may be endowed with a group law. In order for the group law and the topology to interweave well, the group operations must be continuous functions; informally, formula_465 and formula_466 must not vary wildly if formula_186 and formula_189 vary only a little. Such groups are called "topological groups," and they are the group objects in the category of topological spaces. The most basic examples are the group of real numbers under addition and the group of nonzero real numbers under multiplication. Similar examples can be formed from any other topological field, such as the field of complex numbers or the field of -adic numbers. These examples are locally compact, so they have Haar measures and can be studied via harmonic analysis. Other locally compact topological groups include the group of points of an algebraic group over a local field or adele ring; these are basic to number theory Galois groups of infinite algebraic field extensions are equipped with the Krull topology, which plays a role in infinite Galois theory. A generalization used in algebraic geometry is the étale fundamental group.


title: Group (mathematics)

text: A "Lie group" is a group that also has the structure of a differentiable manifold; informally, this means that it looks locally like a Euclidean space of some fixed dimension. Again, the definition requires the additional structure, here the manifold structure, to be compatible: the multiplication and inverse maps are required to be smooth.


title: Group (mathematics)

text: A standard example is the general linear group introduced above: it is an open subset of the space of all formula_321-by-formula_321 matrices, because it is given by the inequality


title: Group (mathematics)

text: Lie groups are of fundamental importance in modern physics: Noether's theorem links continuous symmetries to conserved quantities. Rotation, as well as translations in space and time, are basic symmetries of the laws of mechanics. They can, for instance, be used to construct simple models—imposing, say, axial symmetry on a situation will typically lead to significant simplification in the equations one needs to solve to provide a physical description. Another example is the group of Lorentz transformations, which relate measurements of time and velocity of two observers in motion relative to each other. They can be deduced in a purely group-theoretical way, by expressing the transformations as a rotational symmetry of Minkowski space. The latter serves—in the absence of significant gravitation—as a model of spacetime in special relativity. The full symmetry group of Minkowski space, i.e., including translations, is known as the Poincaré group. By the above, it plays a pivotal role in special relativity and, by implication, for quantum field theories. Symmetries that vary with location are central to the modern description of physical interactions with the help of gauge theory. An important example of a gauge theory is the Standard Model, which describes three of the four known fundamental forces and classifies all known elementary particles.


title: Group (mathematics)

text: More general structures may be defined by relaxing some of the axioms defining a group. The table gives a list of several structures generalizing groups.


title: Group (mathematics)

text: For example, if the requirement that every element has an inverse is eliminated, the resulting algebraic structure is called a monoid. The natural numbers formula_475 (including zero) under addition form a monoid, as do the nonzero integers under multiplication formula_476. Adjoining inverses of all elements of the monoid formula_476 produces a group formula_478, and likewise adjoining inverses to any (abelian) monoid produces a group known as the Grothendieck group of .


title: Group (mathematics)

text: A group can be thought of as a small category with one object in which every morphism is an isomorphism: given such a category, the set formula_479 is a group; conversely, given a group , one can build a small category with one object in which formula_480.


title: Group (mathematics)

text: In a groupoid, the set of all morphisms in the category is usually not a group, because the composition is only partially defined: is defined only when the source of matches the target of .


title: Group (mathematics)

text: Finally, it is possible to generalize any of these concepts by replacing the binary operation with an -ary operation (i.e., an operation taking arguments, for some nonnegative integer ). With the proper generalization of the group axioms, this gives a notion of -ary group.


title: 1815 eruption of Mount Tambora

text: Mount Tambora is a volcano on the island of Sumbawa in present-day Indonesia, then part of the Dutch East Indies, and its 1815 eruption was the most powerful volcanic eruption in recorded human history. This volcanic explosivity index (VEI) 7 eruption ejected of material into the atmosphere, and was the most recent confirmed VEI-7 eruption.


title: 1815 eruption of Mount Tambora

text: Although the Mount Tambora eruption reached a violent climax on 10 April 1815, increased steaming and small phreatic eruptions occurred during the next six months to three years. The ash from the eruption column dispersed around the world and lowered global temperatures in an event sometimes known as the Year Without a Summer in 1816. This brief period of significant climate change triggered extreme weather and harvest failures in many areas around the world. Several climate forcings coincided and interacted in a systematic manner that has not been observed after any other large volcanic eruption since the early Stone Age.


title: 1815 eruption of Mount Tambora

text: Mount Tambora experienced several centuries of dormancy before 1815, caused by the gradual cooling of hydrous magma in its closed magma chamber. Inside the chamber at depths between , the exsolution of a high-pressure fluid magma formed during cooling and crystallisation of the magma. An over-pressurization of the chamber of about was generated, with the temperature ranging from . In 1812, the volcano began to rumble and generated a dark cloud.


title: 1815 eruption of Mount Tambora

text: On 5 April 1815, a giant eruption occurred, followed by thunderous detonation sounds heard in Makassar on Sulawesi away, Batavia (now Jakarta) on Java away, and Ternate on the Molucca Islands away. On the morning of 6 April, volcanic ash began to fall in East Java with faint detonation sounds lasting until 10 April. What was first thought to be the sound of firing guns was heard on 10 April on Sumatra, more than away.


title: 1815 eruption of Mount Tambora

text: At about 19:00 on 10 April, the eruptions intensified. Three plumes rose up and merged. The whole mountain was turned into a flowing mass of "liquid fire". Pumice stones of up to in diameter started to rain down around 20:00, followed by ash at around 21:00–22:00. Pyroclastic flows cascaded down the mountain to the sea on all sides of the peninsula, wiping out the village of Tambora. Loud explosions were heard until the next evening, 11 April. The ash veil spread as far as West Java and South Sulawesi. A nitrous odor was noticeable in Batavia, and heavy tephra-tinged rain fell, finally receding between 11 and 17 April.


title: 1815 eruption of Mount Tambora

text: The explosion had an estimated VEI of 7. An estimated of pyroclastic trachyandesite were ejected, weighing about 10 billion tonnes. This left a caldera measuring across and deep. The density of fallen ash in Makassar was . Before the explosion, Mount Tambora's peak elevation was about , making it one of the tallest peaks in the Indonesian archipelago. After the explosion, its peak elevation had dropped to only , about two-thirds of its previous height.


title: 1815 eruption of Mount Tambora

text: The 1815 Tambora eruption is the largest observed eruption in recorded history, as shown in the table below. The explosion was heard away, and ash fell at least away.


title: 1815 eruption of Mount Tambora

text: All vegetation on the island was destroyed. Uprooted trees, mixed with pumice ash, washed into the sea and formed rafts up to across. Between 1 and 3 October the British ships and "James Sibbald" encountered extensive pumice rafts about west of Tambora. Clouds of thick ash still covered the summit on 23 April. Explosions ceased on 15 July, although smoke emissions were observed as late as 23 August. Flames and rumbling aftershocks were reported in August 1819, four years after the event.


title: 1815 eruption of Mount Tambora

text: A moderate-sized tsunami struck the shores of various islands in the Indonesian archipelago on 10 April, with a height of up to in Sanggar around 22:00. A tsunami of in height was reported in Besuki, East Java, before midnight, and one of in height in the Molucca Islands. The total death toll has been estimated to be around 4,600.


title: 1815 eruption of Mount Tambora

text: The eruption column reached the stratosphere at an altitude of more than . The coarser ash particles settled out one to two weeks after the eruptions, but the finer ash particles stayed in the atmosphere from a few months to a few years at altitudes of . Longitudinal winds spread these fine particles around the globe, creating optical phenomena. Prolonged and brilliantly coloured sunsets and twilights were seen frequently in London between 28 June and 2 July 1815 and 3 September and 7 October 1815. The glow of the twilight sky typically appeared orange or red near the horizon and purple or pink above.


title: 1815 eruption of Mount Tambora

text: The estimated number of deaths varies depending on the source. Zollinger (1855) puts the number of direct deaths at 10,000, probably caused by pyroclastic flows. On Sumbawa, 18,000 starved to death or died of disease. About 10,000 people on Lombok died from disease and hunger. Petroeschevsky (1949) estimated that about 48,000 people were killed on Sumbawa and 44,000 on Lombok. Stothers in 1984 and several other authors have accepted Petroeschevsky's claim of 88,000 deaths in total. However, a 1998 journal article authored by J. Tanguy and others claimed that Petroeschevsky's figures were unfounded and based on untraceable references. Tanguy's revision of the death toll was based on Zollinger's work on Sumbawa for several months after the eruption and on Thomas Raffles's notes. Tanguy pointed out that there may have been additional victims on Bali and East Java because of famine and disease. Their estimate was 11,000 deaths from direct volcanic effects and 49,000 by post-eruption famine and epidemic diseases. Oppenheimer wrote that there were at least 71,000 deaths in total. Reid has estimated that 100,000 people on Sumbawa, Bali, and other locations died from the direct and indirect effects of the eruption.


title: 1815 eruption of Mount Tambora

text: The eruption caused a volcanic winter. During the Northern Hemisphere summer of 1816, global temperatures cooled by . This very significant cooling directly or indirectly caused 90,000 deaths. The eruption of Mount Tambora was the most significant cause of this climate anomaly. While there were other eruptions in 1815, Tambora is classified as a VEI-7 eruption with a column tall, eclipsing all others by at least one order of magnitude.


title: 1815 eruption of Mount Tambora

text: The VEI is used to quantify the amount of ejected material, with a VEI-7 being . Every index value below that is one order of magnitude (meaning, ten times) less. Furthermore, the 1815 eruption occurred during a Dalton Minimum, a period of unusually low solar radiation. Volcanism plays a large role in climate shifts, both locally and globally. This was not always understood and did not enter scientific circles as fact until the 1883 eruption of Krakatoa tinted the skies orange.


title: 1815 eruption of Mount Tambora

text: The scale of the volcanic eruption will determine the significance of the impact on climate and other chemical processes, but a change will be measured even in the most local of environments. When volcanoes erupt, they eject carbon dioxide (CO), water, hydrogen, sulfur dioxide (SO2), hydrogen chloride, hydrogen fluoride, and many other gases.(Meronen et al. 2012) CO and water are greenhouse gases, comprising 0.0415 percent and 0.4 percent of the atmosphere, respectively. Their small proportion disguises their significant role in trapping solar insolation and reradiating it back to Earth.


title: 1815 eruption of Mount Tambora

text: The 1815 eruption released SO into the stratosphere, causing a global climate anomaly. Different methods have estimated the ejected sulphur mass during the eruption: the petrological method; an optical depth measurement based on anatomical observations; and the polar ice core sulfate concentration method, using cores from Greenland and Antarctica. The figures vary depending on the method, ranging from 10 to 120 million tonnes.


title: 1815 eruption of Mount Tambora

text: In the spring and summer of 1815, a persistent "dry fog" was observed in the northeastern United States. The fog reddened and dimmed the sunlight, such that sunspots were visible to the naked eye. Neither wind nor rainfall dispersed the "fog". It was identified as a stratospheric sulfate aerosol veil. In summer 1816, countries in the Northern Hemisphere suffered extreme weather conditions, dubbed the "Year Without a Summer". Average global temperatures decreased by about , enough to cause significant agricultural problems around the globe. On 4 June 1816, frosts were reported in the upper elevations of New Hampshire, Maine, Vermont, and northern New York. On 6 June 1816, snow fell in Albany, New York and Dennysville, Maine. On 8 June 1816, the snow cover in Cabot, Vermont was reported still to be deep. Such conditions occurred for at least three months and ruined most agricultural crops in North America. Canada experienced extreme cold during that summer. Snow deep accumulated near Quebec City from 6 to 10 June 1816.


title: 1815 eruption of Mount Tambora

text: The second-coldest year in the Northern Hemisphere since around 1400 was 1816, and the 1810s are the coldest decade on record. That was the consequence of Tambora's 1815 eruption and possibly another VEI-6 eruption in late 1808. The surface temperature anomalies during the summer of 1816, 1817, and 1818 were , , and , respectively. Parts of Europe also experienced a stormier winter.


title: 1815 eruption of Mount Tambora

text: This climate anomaly has been blamed for the severity of typhus epidemics in southeast Europe and along the eastern Mediterranean Sea between 1816 and 1819. The climate changes disrupted the Indian monsoons, caused three failed harvests and famine, and contributed to the spread of a new strain of cholera that originated in Bengal in 1816. Many livestock died in New England during the winter of 1816–1817. Cool temperatures and heavy rains resulted in failed harvests in the British Isles. Families in Wales travelled long distances as refugees, begging for food. Famine was prevalent in north and southwest Ireland, following the failure of wheat, oat, and potato harvests. The crisis was severe in Germany, where food prices rose sharply, and demonstrations in front of grain markets and bakeries, followed by riots, arson, and looting, took place in many European cities. It was the worst famine of the 19th century.


title: 1815 eruption of Mount Tambora

text: Volcanism affects the atmosphere in two distinct ways: short-term cooling caused by reflected insolation and long-term warming from increased CO levels. Most of the water vapor and CO is collected in clouds within a few weeks to months because both are already present in large quantities, so the effects are limited. It has been suggested that a volcanic eruption in 1809 may also have contributed to a reduction in global temperatures.


title: 1815 eruption of Mount Tambora

text: By most calculations, the eruption of Tambora was at least a full order of magnitude (10 times) larger than that of Mount Pinatubo in 1991.(Graft et al. 1993) An estimated of the top of the mountain collapsed to form a caldera, reducing the height of the summit by a third. Around of rock was blasted into the air.(Williams 2012) Toxic gases also were pumped into the atmosphere, including sulfur that caused lung infections.(Cole-Dai et al. 2009) Volcanic ash was over deep within of the eruption, while areas within a radius saw a ash fall, and ash could be found as far away as . The ash burned and smothered crops, creating an immediate shortage of food in Indonesia.(Cole-Dai et al. 2009) The ejection of these gases, especially hydrogen chloride, caused the precipitation to be extremely acidic, killing much of the crops that survived or were rebudding during the spring. The food shortages were compounded by the Napoleonic wars, floods, and cholera. Its energy release was equivalent to about .


title: 1815 eruption of Mount Tambora

text: The ash in the atmosphere for several months after the eruption reflected significant amounts of solar radiation, causing unseasonably cool summers that contributed to food shortages. China, Europe, and North America had well-documented below normal temperatures, which devastated their harvests. The monsoon season in China and India was altered, causing flooding in the Yangtze Valley and forcing thousands of Chinese to flee coastal areas. (Granados et al. 2012) The gases also reflected some of the already-decreased incoming solar radiation, causing a decrease in global temperatures throughout the decade. An ice dam formed in Switzerland during the summers of 1816 and 1817, earning 1816 the title "Year without a Summer". The winter months of 1816 were not much different from previous years, but the spring and summer maintained the cool-to-freezing temperatures. The winter of 1817, however, was radically different, with temperatures below in central and northern New York, which were cold enough to freeze lakes and rivers that were normally used to transport supplies. Both Europe and North America suffered from freezes lasting well into June, with snow accumulating to in August, which killed recently planted crops and crippled the food industry. The length of the growing seasons in parts of Massachusetts and New Hampshire were less than 80 days in 1816, resulting in harvest failures (Oppenheimer 2003). Visually unique sunsets were observed in western Europe, and red fog was observed along the eastern coast of the U.S. These unique atmospheric conditions persisted for the better part of 2.5 years.(Robock 2000)


title: 1815 eruption of Mount Tambora

text: Scientists have used ice cores to monitor atmospheric gases during the cold decade (1810–1819), and the results have been puzzling. The sulfate concentrations found in both Siple Station, Antarctica and central Greenland bounced from 5.0 in January 1816 to 1.1 in August 1818. This means that 25–30 teragrams of sulfur were ejected into the atmosphere, most of which came from Tambora, followed by a rapid decrease through natural processes. Tambora caused the largest shift in sulfur concentrations in ice cores for the past 5,000 years. Estimates of the sulfur yield vary from 10 teragrams (Black et al. 2012) to 120 teragrams,(Stothers 2000) with the average of the estimates being 25–30 teragrams. The high concentrations of sulfur could have caused a four-year stratospheric warming of around , resulting in a delayed cooling of surface temperatures that lasted for nine years.(Cole-Dai et al. 2009) This has been dubbed a "volcanic winter", similar to a nuclear winter because of the overall decrease in temperatures and abysmal farming conditions.


title: 1815 eruption of Mount Tambora

text: Climate data have shown that the variance between daily lows and highs may have played a role in the lower average temperature because the fluctuations were much more subdued. Generally, the mornings were warmer because of nightly cloud cover and the evenings were cooler because the clouds had dissipated. There were documented fluctuations of cloud cover for various locations that suggested it was a nightly occurrence and the sun killed them off, much like a fog. The class boundaries between 1810–1830 without volcanically perturbed years was around . In contrast, the volcanically perturbed years (1815–1817) had a change of only around . This meant that the mean annual cycle in 1816 was more linear than bell shaped and 1817 endured cooling across the board. Southeastern England, northern France, and the Netherlands experienced the greatest amount of cooling in Europe, accompanied by New York, New Hampshire, Delaware, and Rhode Island in North America.


title: 1815 eruption of Mount Tambora

text: The documented rainfall was as much as 80 percent more than the calculated normal with regards to 1816, with unusually high amounts of snow in Switzerland, France, Germany, and Poland. This is again contrasted by the unusually low precipitations in 1818, which caused droughts throughout most of Europe and Asia.(Auchmann et al. 2012) Russia had already experienced unseasonably warm and dry summers since 1815 and this continued for the next three years. There are also documented reductions in ocean temperature near the Baltic Sea, the North Sea, and the Mediterranean Sea. This seems to have been an indicator of shifted oceanic circulation patterns and possibly changed wind direction and speed.(Meronen et al. 2012)


title: 1815 eruption of Mount Tambora

text: Taking into account the Dalton Minimum and the presence of famine and droughts predating the eruption, the Tambora eruption accelerated or exacerbated the extreme climate conditions of 1815. While other eruptions and other climatological events would have led to a global cooling of about


title: Transgender history

text: Transgender people (including non-binary and third gender individuals) have existed in cultures worldwide since ancient times. The modern terms and meanings of "transgender", "gender", "gender identity", and "gender role" only emerged in the 1950s and 1960s. As a result, opinions vary on how to categorize historical accounts of gender-variant people and identities.


title: Transgender history

text: Sumerian and Akkadian texts from 4,500 years ago document priests known as "gala" who may have been transgender. Likely depictions occur in art around the Mediterranean from 9,000 to 3,700 years ago. In Ancient Greece, Phrygia, and Rome, there were "galli" priests that some scholars believe to have been trans women. Roman emperor Elagabalus (d. 222 AD) preferred to be called a lady (rather than a lord) and sought sex reassignment surgery, and in the modern day has been seen as a trans figure. "Hijras" on the Indian subcontinent and "kathoeys" in Thailand have formed trans-feminine third gender social and spiritual communities since ancient times, with their presence documented for thousands of years in texts which also mention trans male figures. Today, at least half a million "hijras" live in India and another half million in Bangladesh, legally recognized as a third gender, and many trans people are accepted in Thailand. In Arabia, "khanith" today (like earlier "mukhannathun") fulfill a third gender role attested since the AD 600s. In Africa, many societies have traditional roles for trans women and trans men, some of which survive in the modern era. In the Americas prior to European colonization, as well as in some contemporary North American Indigenous cultures, there are social and ceremonial roles for third gender people, or those whose gender expression transforms, such as the Navajo "nádleehi" or the Zuni "lhamana".


title: Transgender history

text: In the Middle Ages, accounts around Europe document transgender people. Kalonymus ben Kalonymus's lament for being born a man instead of a woman has been seen as an early account of gender dysphoria. Eleanor Rykener, a male-bodied Briton arrested in 1394 while living and doing sex work dressed as a woman, has been seen as a trans woman. In the Balkans since the 1400s, female-assigned people have transitioned to live as men called sworn virgins. In Japan, accounts of trans people go back to the Edo period. In Indonesia, there are millions of trans-/third-gender "waria", and the "bugis" of Sulawesi recognize five genders. In Oceania, trans-/third-gender roles like the "akava'ine", "fa'afafine" and "fakaleiti" exist among the Cook Island Maori, Samoans, and Tongans.


title: Transgender history

text: In colonial America, Thomas(ine) Hall in the 1600s adopted clothes and roles of both men and women, while in 1776 the genderless Public Universal Friend refused both birth name and gendered pronouns. During the 1800s, some people began new lives as men and served in the military, including Albert Cashier and James Barry, or otherwise transitioned, like Joseph Lobdell; trans women like Frances Thompson also transitioned. In 1895, trans autobiographer Jennie June and others organized the Cercle Hermaphroditos; in the 1900s, musician Billy Tipton lived as a man, while Lucy Hicks Anderson was supported by her parents and community in being a woman.


title: Transgender history

text: Karl M. Baer (in 1906) and Alan L. Hart (1917) underwent early female-to-male reassignment surgeries, while in 1930 and 1931 Dora Richter and Lili Elbe had early male-to-female surgeries including (for Elbe) an ovary and uterus transplant. Baer, Richter and Elbe were aided by Magnus Hirschfeld, whose pioneering work at the Institut für Sexualwissenschaft for trans medicine and rights was destroyed by the Nazis in 1933. In 1952, American trans woman Christine Jorgensen's transition brought wide awareness of sex reassignment surgery to North America, while Coccinelle's 1958 transition did the same in Europe. The grassroots fight for trans rights became more visible with trans and gay people fighting back against police in the 1959 Cooper Donuts Riot, 1966 Compton's Cafeteria Riot, and multi-day Stonewall Riots of 1969. In the 1970s, Lou Sullivan pioneered visibility for gay trans men and organized what became FTM International. At the same time, some feminists opposed the inclusion of trans women, creating what was later known as trans-exclusionary radical feminism. In the 1990s and 2000s, the Transgender Day of Remembrance was started and trans marches became more common; trans people were elected to public offices; and legislative and court actions began recognizing trans people's rights in some countries (especially in the West, India, and southern Africa). At the same time, other countries (especially in the rest of Africa, Central Asia, and Arabia) abridge trans people's rights due to transphobia.


title: Transgender history

text: Transgender people are known to have existed since ancient times. A wide range of societies had traditional third gender roles, or otherwise accepted trans people in some form. However, a precise history is difficult because the modern concept of being transgender, and gender in general, did not develop until the mid-1900s. Historical understandings are thus inherently filtered through modern principles, and were largely viewed through a medical lens until the late 1900s. Writer Genny Beemyn points out:


title: Transgender history

text: Trans history has also been filtered through gay history, with some historians erasing the trans identities of historical figures such as Billy Tipton to instead promote a gay reading of their lives.


title: Transgender history

text: The absence of autobiographical accounts requires historians to assign identities to historical figures, which of course may be inaccurate. Author Jason Cromwell assesses that if a person indicated he was a man, modified his body to look more traditionally male, and lived his life as a man, he was a trans man; the same approach has been used to identify trans women. Genny Beemyn distinguishes trans people from crossdressers in the historical record by assessing that a person who crossdressed only in public did not mind exposing their dual life as a crossdresser, while those who crossdressed consistently (also in private) and sought to keep their assigned gender a secret were more likely trans.


title: Transgender history

text: Beemyn also distinguishes non-binary individuals in the historical record. She notes that many societies in the New World had non-binary gender roles enshrined in their society, which enraged European explorers. For example, in 1513 Vasco Núñez de Balboa killed 40 natives on the Panama Isthmus for being sodomites, as they had been assigned male at birth but were practicing female gender roles. Not all Europeans were as judgmental: a matter-of-fact 1564 narrative describes "hermaphrodites" as "quite common". An account from Edwin Thompson Denig in the first half of the 19th century describes a "neuter" gender among the Crow people. Denig said of it: "Strange country this, where males assume the dress and perform the duties of females, while women turn men and mate with their own sex!" Beemyn concludes that European writers lacked the language or cultural understanding to adequately describe the practices they were witnessing. Overall, she cautions not to make generalizations about native practices, since third gender roles were extremely diverse and ranged from exalted positions who were believed to have supernatural power, to denigrated underlings.


title: Transgender history

text: Ancient Egypt had third gender categories, including for eunuchs. In the Tale of Two Brothers (from 3,200 years ago), Bata removes his penis and tells his wife "I am a woman just like you"; one modern scholar called him temporarily (before his body is restored) "transgendered". Mut, Sekhmet and other goddesses are sometimes represented androgynously, with erect penises, and Anat wears clothes of both men and women.


title: Transgender history

text: Trans people face stigma and are not able to change gender markers or access hormone therapy or reassignment surgery in Morocco, but in 2018 some founded a group to oppose discrimination. In Algeria, trans people mostly live in the shadows, or seek refuge in France; in 2014 the first LGBT magazine in the country, "El Shad", launched and profiled several. In Tunisia, trans people have been arrested, jailed, and tortured; some seek asylum in Greece. Egypt today is also hostile to transgender people, who are subject to arrest.


title: Transgender history

text: The Nuba peoples of Sudan (including the Otoro Nuba, Nyima, Tira, Krongo, and Mesakin), have traditional roles for male-assigned people who dress and live as women and may marry men, which have been seen as transgender roles. However, trans people face discrimination in the modern Sudanese state, and cross-dressing is illegal.


title: Transgender history

text: By the modern period, the Igbo, like many other peoples, had gender and transgender roles, including for females who take on male status and marry women, a practice which also exists among the Dahomey (Fon) of Benin and has been viewed through both transgender and homosexual lenses. Anthropologist John McCall documented a female-assigned Ohafia Igbo named Nne Uko Uma Awa, who dressed and behaved as a boy since childhood, joined men's groups, and was a husband to two wives; in 1991, Awa stated "by creation I was meant to be a man. But as it happened, when coming into this world I came with a woman's body. That is why I dressed [as a man]." However, trans people in Nigeria face harassment and violence.


title: Transgender history

text: In the modern Ghanaian state, trans people face violence and discrimination in accessing healthcare, work, education and housing, as they also do in a number of other western African states like the Gambia.


title: Transgender history

text: Trans people face abuse from society, government, media and doctors in Senegal, and are harassed (including by police) in Sierra Leone, but have built some underground community spaces. Transphobia is rampant in modern Mali and trans women are often beaten in the streets. In Liberia, sexual minorities have long been part of society, and founded the Transgender Network of Liberia in 2014, hold an annual pageant, and mark the Trans Day of Remembrance, but also face harassment. They benefited from US backing under Obama and were harmed by Trump administration cuts, and by Liberians who wrongly believe transness was introduced to the country by the West.


title: Transgender history

text: In the Ivory Coast, trans women (especially sex workers) face harassment and violence, especially since the 2011 election; since 2009, there has been an annual drag pageant, but it focuses more on gay men than trans women or "travestis". In modern Benin, one trans woman was supported by her mother and the French in organizing other trans Beninese, but abused by other relatives, threatened by police, and forced to flee abroad. In Cape Verde, activist Tchinda Andrade came out in 1998, becoming so well known that trans people are locally called "tchindas"; in 2015, the documentary "Tchindas" followed her preparation for the annual carnival. Trans people still face intolerance, but São Vicente, Cape Verde is today among the more tolerant places in Africa, which locals attribute to its small size requiring people to work together.


title: Transgender history

text: In Cameroon, trans people face violence and discrimination in accessing healthcare, work, education and housing, and trans women have been attacked and jailed. Trans people in the Democratic Republic of the Congo today also face harassment. Trans and gay people in Rwanda live more openly and face less violence than in neighboring states, but face some stigma. In Angola, in the 2010s, trans singer Titica initially faced violence but has become popular, especially with young Angolans.


title: Transgender history

text: Among Swahili-speaking people in Kenya, male-assigned may take feminine names, marry men, and do womanly household work (while marry women). Among some other Kenyan peoples, male-assigned priests (called "mugawe" among the Meru and Kikuyu) dress and style their hair like women and may marry men, and have been compared to trans women.


title: Transgender history

text: Among the Nuer people (in what is now South Sudan and Ethiopia), female-assigned people who have borne no children may adopt a male status, marry a woman, and be regarded as the father of any children they bear (a practice which has been viewed as transgender or homosexual); the Nuer are also reported to have a male-to-female role. The Maale people of Ethiopia also have a traditional role for male-assigned "ashtime" who take on feminine roles; traditionally, they served as sexual partners for the king on days he was ritually barred from sex with women; with the introduction of modern transphobia, "ashtime" came to be viewed as abnormal by the 1970s. The Amhara people of Ethiopia stigmatize male-assigned people in their communities who adopt feminine dress.


title: Transgender history

text: In Uganda today, transphobia and homophobia is increasing, introduced in the 1800s and 1900s by Christian missionaries and stoked in the 2000s by conservative evangelicals; trans people are now often kicked out by their families and denied work, and face discrimination in accessing healthcare, though trans men are trying to challenge such transphobia and sexist gender roles. Traditionally, Ugandan peoples were largely accepting of trans and gay people; the Lango people accepted trans women—male-assigned people called "jo apele" or "jo aboich" who were believed to have been transformed at conception into women by the androgynous deity Jok, and who adopted women's names, dress, and face-decorations, grew their hair long, simulated menstruation, and could marry men—as did the Karamojong and Teso, and the Lugbara people had roles for both trans women () and trans men ().


title: Transgender history

text: In Madagascar, the U.S. State Department reported in 2011 that "sexual orientation and gender identity were not widely discussed" and attitudes ranged "from tacit acceptance to violent rejection, particularly of transgender sex workers". In the early 2000s, Balou Chabart Rasoana became one of the first publicly out trans women, and faced discrimination but was supported by her mother and, over time, her neighborhood; much of the LGBT community remains underground.


title: Transgender history

text: Various Bantu peoples in southern Africa, including the Zulu, Basotho, Mpondo and Tsonga, had a tradition of young men ("inkotshane" in Zulu, "boukonchana" in Sesotho, "tinkonkana" in Mpondo, and "nkhonsthana" in Tsonga; called "boy-wives" in English) who married or had intercrural or anal sex with older men, and sometimes dressed as women, wore breast prostheses, did not grow beards, and did women's work; these relationships became common among South African miners and continued into the 1950s, and while often interpreted as homosexual, boy-wives are sometimes seen as transgender.


title: Transgender history

text: In two cases in 2017, Botswana's High Court ruled trans men and trans women have the right to have their gender identity recognized by the government and to change gender markers; the court said the registrar's refusal to change a marker was unreasonable and violated the person's "rights to dignity, privacy, freedom of expression, equal protection of the law, freedom from discrimination and freedom from inhumane and degrading treatment".


title: Transgender history

text: From the 1960s to 1980s, the South African Defence Force forced some white gay and lesbian soldiers to have sex reassignment surgery.


title: Transgender history

text: Since March 2004, trans and intersex people are allowed to change their legal sex after medical treatment such as hormone replacement therapy. Several Labour Court rulings have found against employers that mistreated employees who transitioned.


title: Transgender history

text: Prior to western contact, some Native American tribes had third-gender roles, like the Diné (Navajo) "nádleehi" and the Zuni "lhamana". European anthropologists usually referred to these people as "berdaches", which Indigenous people have always considered an offensive slur. In 1990, some Indigenous North Americans, largely in academia, adopted the pan-Indian neologism "two-spirit", as an attempt to organize inter-tribally. Though acceptance of this term in traditional Native communities which already have their own terms for such people has been limited, it has generally met with more acceptance than the slur it replaced.


title: Transgender history

text: One of the first European accounts of Iroquois practices of gender was made by missionary Joseph-François Lafitau who spent six years among the Iroquois starting in 1711, and observed "women with manly courage who prided themselves upon the profession of warrior, [and seemed] to become men alone", and people he called "men cowardly enough to live as women."


title: Transgender history

text: There is archaeological evidence that trans- or third-gender individuals existed in California 2,500 years ago at rates comparable to those at which they exist among indigenous peoples there in the modern era, and archaeological and ethnographic evidence suggests third-gender categories may be of great antiquity in North America overall; Barbara Voss suggests they may go back to the first migrations of people from eastern Asia and Siberia over 10,000 years ago.


title: Transgender history

text: During the colonial period a European system of beliefs and values was imposed on the First Nations and enforced among the colonists.


title: Transgender history

text: In 1970, Dianna Boileau underwent sex reassignment surgery at Toronto General Hospital, becoming possibly the first in Canada to do so. Over the following two years, Boileau shared her story with a number of press outlets and published a 1972 memoir, "Behold, I Am a Woman", before retreating from the public eye.


title: Transgender history

text: In 2002, sexual orientation and gender identity were included in the Northwest Territories Human Rights Act.


title: Transgender history

text: In June 2012, gender identity and expression were added to the Ontario Human Rights Code, and gender identity was added to the Manitoba Human Rights Code. In December 2012 Nova Scotia added gender identity and expression to the list of things explicitly protected from harassment in that province's Human Rights Act. In May 2012, after a legal battle to reverse her disqualification for not being a "naturally born female", Vancouver resident Jenna Talackova became the first trans woman to compete in a Miss Universe pageant, and was one of four contestants to win "Miss Congeniality".


title: Transgender history

text: In March 2013, the House of Commons passed Bill C-279 to officially extend human rights protections to trans people in Canada. In February 2015, the Senate of Canada amended the bill in ways that were criticized as transphobic.


title: Transgender history

text: In December 2015, legislator Estefania Cortes-Vargas came out as non-binary in the Legislative Assembly of Alberta during a debate over the inclusion of transgender rights in the provincial human rights code. While the provincial "Hansard" normally reports members' speeches under the gender honorifics "Mr." or "Ms.", Cortes-Vargas is recorded as "Member Cortes-Vargas". On December 17, 2015, Kael McKenzie was appointed to the Provincial Court of Manitoba, becoming Canada's first openly transgender judge.


title: Transgender history

text: In 2016, gender identity or expression was added to the Quebec Charter of Rights and Freedoms. The same year, Jennifer Pritzker gave a $2 million donation to create the world's first endowed academic chair of transgender studies, at the University of Victoria in British Columbia; Aaron Devor was chosen as the inaugural chair. In May 2016, Bill C-16 was introduced aiming to update the Canadian Human Rights Act and Criminal Code to include gender identity and expression as protected grounds from discrimination, hate publication and advocacy of genocide, and to add targeting of victims on the basis of gender identity and expression to the list of aggravating factors in sentencing, the first time such a bill was put forward by the governing party in the House of Commons. Since June 2017, all places within Canada explicitly within the Canadian Human Rights Act or equal opportunity or anti-discrimination legislation do prohibit discrimination against gender identity or expression.


title: Transgender history

text: Since August 2017, Canadians can indicate that they are neither male nor female on their passports, using an 'x' marker.


title: Transgender history

text: In January 2018, Canadian Women's Hockey League player Jessica Platt came out, the first trans woman to come out in North American professional hockey.


title: Transgender history

text: In 1791, early in the Haitian Revolution, a black planter who had been raised as a boy led an uprising in southern Haiti under the name Romaine-la-Prophétesse ("Romaine the Prophetess"). Romaine dressed like a woman and spoke of being possessed by a female spirit, may have been transgender or genderfluid, and has been compared to the transgender feminine religious figures of West Africa, the area many black Haitians descended from. Mary Grace Albanese and list Romaine among the women who led the Haitian Revolution, while Terry Rey argues calling Romaine transgender could be anachronistic. Romaine has been compared to Kimpa Vita, who professed to be the incarnation of a male Catholic saint.


title: Transgender history

text: In the modern era, discrimination and violence against transgender people is common in Haitian society, though many LGBT people find it easier to be open about their gender within the Vodou subculture, in which it is believed, for example, that people may be possessed by divinities of the opposite sex. Haiti's criminal code prohibits vagrancy, with a specific mention of transvestites.


title: Transgender history

text: In several pre-Columbian communities across Mexico, anthropologists and colonial accounts document acceptance of third-gender categories. Transvestitism was an accepted practice in the native cultures of Central (and South) America, including among the Aztecs and Mayans (as reflected in their mythologies). Spanish colonizers were hostile to it.


title: Transgender history

text: The Zapotec people of Oaxaca have a third gender role for "muxes", people who dress, behave and perform work otherwise associated with the other binary gender; "vestidas" wear feminine clothes, while "pintadas" wear masculine clothes but also makeup and jewellery. They may marry women, men, or other muxes. It has been suggested that while the three gender system predates Spanish colonization, the phenomenon of muxes dressing as women may be more recent. Juchitán de Zaragoza, an indigenous community on the Isthmus of Tehuantepec, has so many well-accepted muxes there is a myth attributing their numbers to a bag of third-genders carried by Saint Vicent ripping and accidentally spilling many out over the town; one study estimated 6% of males in the community in the 1970s were muxes.


title: Transgender history

text: During the Mexican Revolution, Amelio Robles Ávila began to dress and demand to be treated as a man and, gaining respect as a capable leader, was promoted to colonel. Robles' maleness was accepted by family, society, and the Mexican government, and he lived as a man from age 24 until death; a neighbor said that if anyone called Robles a woman, Robles would threaten them with a pistol, and he killed two men who attacked him and tried to reveal his anatomy.


title: Transgender history

text: Thomas(ine) Hall, an indentured servant in Virginia, reported being both a man and a woman and adopted clothes and roles of each at different times until ordered by a court in 1629 to wear both men's breeches and a woman's apron; Hall is thought to have been intersex and is cited as an early example of "a gender nonconforming individual in colonial America".


title: Transgender history

text: In 1776, the Public Universal Friend reported being genderless, dressed androgynously, and asked followers gained while preaching throughout New England over the next four decades not to use their birth name or gendered pronouns; some scholars have called the Friend a chapter in trans history "before [the word] 'transgender'". There were also cases of people living as the opposite gender in the early years of the Republic, such as Joseph Lobdell, who was assigned female at birth in 1829, lived as a man for sixty years, and married a woman. Charley Parkhurst was a stagecoach driver who was assigned female at birth but lived his professional life as a man.


title: Transgender history

text: During the Civil War, over 200 people who had been assigned female at birth donned men's clothing and fought as soldiers; some lived the rest of their lives as men and are thought by some to have been transgender, such as Albert Cashier. After the war, Frances Thompson, a formerly enslaved black trans woman, testified before Congress's investigation of the Memphis Riots of 1866; ten years later, she was arrested for "being a man dressed in women's clothing".


title: Transgender history

text: In the late 1800s, We'wha, a Zuni "lhamana" fiber artist and potter, became a prominent cultural ambassador, visiting Washington, D.C. in 1896 and meeting President Grover Cleveland. The "lhamana" are male-bodied people who may at times take on the social and ceremonial roles usually performed by women in their culture, and at other times the roles more traditionally associated with men.


title: Transgender history

text: In 1895 a group of self-described androgynes in New York organized a club called the Cercle Hermaphroditos, "to unite for defense against the world's bitter persecution". They included Jennie June (assigned male at birth in 1874), whose "The Autobiography of an Androgyne" (1918) was one of a few first-person accounts in the early years of the 20th century which cast light on what life for a transgender person was like then.


title: Transgender history

text: In some cases, immigrants would change their gender identity upon arrival in the United States, especially those assigned female at birth, ostensibly for social mobility, like Frank Woodhull, a Canadian immigrant who lived for about 15 years as a man in California and in 1908 was forced to disclose this during processing at Ellis Island.


title: Transgender history

text: American jazz musician and bandleader Billy Tipton (assigned female at birth in 1914) lived as a man from the 1940s until his death, while socialite and chef Lucy Hicks Anderson insisted as a child that she was a girl and was supported by her parents and doctors and later by the Oxnard, California community in which she was a popular hostess from the 1920s to 1940s. In 1917, Alan L. Hart was one of the first trans men to undergo a hysterectomy and gonadectomy, and later became a pioneering physician and radiologist.


title: Transgender history

text: The possibility of someone changing sex became widely known when Christine Jorgensen in 1952 became the first person widely publicized as undergoing sex reassignment surgery. Around the same time, organizations and clubs began to form, such as Virginia Prince's "Transvestia" publication for an international organization of cross-dressers, but this operated in the same shadows as the still forming gay subculture. In the late 1950s and 1960s, modern transgender and gay activism began with the 1959 Cooper Donuts Riot in Los Angeles, 1966 Compton's Cafeteria riot in San Francisco, and a defining event in gay and transgender activism, the 1969 Stonewall Riots in New York; prominent activists included Sylvia Rivera.


title: Transgender history

text: The 1970s and 1980s saw organizations devoted to transgender social activities or activism come and go, including activist Lou Sullivan's FTM support group that grew into FTM International, the leading advocacy group for trans men. Some feminist and lesbian organizations and individuals began to debate whether transgender women should be accepted into women's groups and events, such as the women's music collective Olivia Records where trans woman Sandy Stone had long been employed, or the Michigan Womyn's Music Festival which had a "women-born-women" in policy.


title: Transgender history

text: The 1990s saw the establishment of Transgender Day of Remembrance to honor those lost to violence, "Paris is Burning" documenting gay and trans New York ball culture, transgender marches and parades around the time of Pride celebrations, and—increasingly in the 2000s and after—the visibility of transgender people rose, with Monica Roberts starting "TransGriot" in the mid-2000s to model accurate media coverage of the trans community, actress Laverne Cox being on the cover of "TIME" in 2014 and Caitlyn Jenner coming out in 2015. Early trans officials like Joanne Conte (elected in 1991 to Arvada, Colorado's city Council) and Althea Garrison (elected to the Massachusetts house in 1992, serving from 1993 to 1995) were not out when elected in the 1990s; while Kim Coco Iwamoto became the first openly trans person elected to statewide office when she won election to the Hawaii Board of Education in 2006 (and later to the Hawaii Civil Rights Commission in 2012), and Danica Roem became the first openly trans person to serve in a state legislature when she won a seat in the Virginia house in 2017.


title: Transgender history

text: Organizations such as the Girl Scouts and the Episcopal Church announced acceptance of transgender members in the 2010s. In 2016, the Obama administration issued guidance that clarified Title IX protections for transgender students, the most well-known being allowing trans students to use bathrooms and locker rooms matching their gender identity. However, some legislative bodies passed discriminatory bills, such as North Carolina's HB 2 (in 2016), and beginning 2017 the Trump administration rescinded the Obama-era protections of trans students, rescinded rules against healthcare providers discriminating against trans patients, and issued a series of orders against employment of trans people by the department of defense. In 2020, the Supreme Court ruled in "Bostock v. Clayton County" that Title VII of the Civil Rights Act protects employees against discrimination because of gender identity (or sexual orientation).


title: Transgender history

text: In 2016, Bolivia passed the Gender Identity Law, which allowed people over 18 to change their name, gender, and picture on legal documents.


title: Transgender history

text: In March 1973, the first sexual reassignment surgery in Latin America took place in Chile, when Marcia Torres underwent it in a Santiago hospital. This took place just months before the 1973 Chilean coup d'état, and the new dictatorship under Augusto Pinochet began adopting policies which criminalized and marginalized the activities of gay and trans people. Torres, however, was able to acquire the changed identity documents she sought from the courts after her surgery.


title: Transgender history

text: In 2018, President Sebastián Piñera signed the Gender Identity Law, which allows transgender people over age 14 "to update their names on legal documents and guarantees their right to be officially addressed according to their true gender."


title: Transgender history

text: In December 2018, Davinson Stiven Erazo Sánchez was charged with the murder of Anyela Ramos Claros, a transgender woman, as a gender-based hate crime. Under the Rosa Elvira Cely law, feminicide, defined as "the killing of a woman because of her gender, or where there were previous instances of violence between the victim and the accused, including sexual violence," was made punishable by a prison sentence of 20 to 50 years. Claros was only the second transgender woman to have her murderer punished under this law.


title: Transgender history

text: Prior to the 16th century arrival of Spanish conquistadors, the Inca Empire and their Moche predecessors revered third-gender persons and organized their society around an Andean cosmovision that made room for masculine and feminine ambiguity based in "complementary dualism." Third-gender shamans as ritual practitioners were subject to violence as the Spanish suppressed pre-colonial worldviews.


title: Transgender history

text: In 2014, the Peruvian Constitutional Court ruled against a transgender woman changing her gender on her national identity document, but in October 2016 the court reversed the earlier decision, acknowledging "people are not only defined by their biological sex, but one must also take into consideration their psychic and social reality." Following this, trans people in Peru can apply to a judge for a gender change without undergoing sex reassignment surgery.


title: Transgender history

text: In 2018, Uruguay passed a law granting rights to transgender people, giving them the right to sex reassignment surgery and hormones paid for by the Uruguayan state. The law also mandates that a minimum number of transgender people be given public jobs. Transgender people can now self-identify and change their legal names without needing approval from a judge. In addition, transgender people who faced persecution during the 1973 to 1985 military dictatorship will receive compensation. The law also lets people under 18 legally change names without the previous requirement of parents' or a court's approval.


title: Transgender history

text: In Sumer, androgynous trans priests known as "gala" used a women's-speech dialect called "eme-sal" and sometimes took female names. During the Akkadian period, similar people known as "kurgarrū" and "assinnu" served Ishtar wearing feminine clothing and performing dances in her temples; the goddess was believed to transform them from masculine to feminine.


title: Transgender history

text: In ancient Assyria, transgender cult prostitutes took part in public processions, singing, dancing, wearing costumes and sometimes women's clothes, carrying feminine symbols, and even at times performing the act of giving birth.


title: Transgender history

text: "Khanith" are a gender category in Oman and Arabia who function in some sexual and social ways as women, and are variously considered to fill an "alternative gender role", to be transgender, or (as they are still considered men by Omani standards and laws) to be transvestites. Discussing the (male-assigned) khanith, older mukhannathun and Egyptian khawalat, and the (female-assigned) ghulamiyat, Everett Rowson writes there is "considerable evidence for institutionalized cross-dressing and other cross-gender behavior in pre-modern Muslim societies, among both men and to some extent women" which existed from Muhammad's day and continued into the Umayyad and Abbasid periods and, in the khanith, into the present.


title: Transgender history

text: Under Mohammad Reza Pahlavi, transsexuals and crossdressers were classed with gays and lesbians and faced lashing or death.