List of transitive finite linear groups

In mathematics, especially in areas of abstract algebra and finite geometry, the list of transitive finite linear groups is an important classification of certain highly symmetric actions of finite groups on vector spaces.
The solvable finite 2-transitive groups were classified by Bertram Huppert. The classification of finite simple groups made possible the complete classification of finite doubly transitive permutation groups. This is a result by Christoph Hering. A finite 2-transitive group has a socle that is either a vector space over a finite field or a non-abelian primitive simple group; groups of the latter kind are almost simple groups and described elsewhere. This article provides a complete list of the finite 2-transitive groups whose socle is elementary abelian.
Let formula_1 be a prime, and formula_2 a subgroup of the general linear group formula_3 acting transitively on the nonzero vectors of the "d"-dimensional vector space formula_4 over the finite fieldformula_5 with "p" elements.
Infinite classes.
There are four infinite classes of finite transitive linear groups.
Notice that the exceptional group of Lie type "G"2("q") is usually constructed as the automorphism groups of the split octonions. Hence, it has a natural representation as a subgroup of the 7-dimensional orthogonal group O(7, "q"). If "q" is even, then the underlying quadratic form polarizes to a degenerate symplectic form. Factoring out with the radical, one obtains an isomorphism between O(7, "q") and the symplectic group Sp(6, "q"). The subgroup of Sp(6, "q") which corresponds to "G"2("q")′ is transitive.
In fact, for "q">2, the group "G"2("q") = "G"2("q")′ is simple. If "q"=2 then "G"2(2)′ ≅ PSU(3,3) is simple with index 2 in "G"2(2).
Sporadic finite transitive linear groups.
These groups are usually classified by some typical normal subgroup, this normal subgroup is denoted by "G"0 and are written in the third column of the table. The notation 21+4− stands for the extraspecial group of minus type of order 32 (i.e. the extraspecial group of order 32 with an odd number (namely one) of quaternion factor).
All but one of the sporadic transitive linear groups formula_2 yield a primitive permutation group formula_11 of degree at most 2499. In the computer algebra programs GAP and MAGMA, these groups can be accessed with the command codice_1 where the number "k" is the "primitive identification" of formula_11. This number is given in the last column of the following table.
Seven of these groups are sharply transitive; these groups were found by Hans Zassenhaus and are also known as the multiplicative groups of the Zassenhaus near-fields. These groups are marked by a star in the table.
This list is not explicitly contained in Hering's paper. Many books and papers give a list of these groups, some of them an incomplete one. For example, Cameron's book misses the groups in line 11 of the table, that is, containing formula_13 as a normal subgroup.


Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field formula_1 of the field of rational numbers such that the field extension formula_2 has finite degree (and hence is an algebraic field extension).
Thus formula_1 is a field that contains formula_4 and has finite dimension when considered as a vector space over
The study of algebraic number fields, and, more generally, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory.
Definition.
Prerequisites.
The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. A prominent example of a field is the field of rational numbers, commonly denoted together with its usual operations of addition and multiplication.
Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)
whose entries are elements of a fixed field, such as the field Any two such sequences can be added by adding the entries one per one. Furthermore, any sequence can be multiplied by a single element "c" of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces are of infinite length. If, however, the vector space consists of "finite" sequences
the vector space is said to be of finite dimension, "n".
Definition.
An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over
Algebraicity and ring of integers.
Generally, in abstract algebra, a field extension formula_27 is algebraic if every element formula_28 of the bigger field formula_1 is the zero of a polynomial with coefficients formula_30 in
Every field extension of finite degree is algebraic. (Proof: for formula_32 in simply consider formula_33 – we get a linear dependence, i.e. a polynomial that formula_32 is a root of.) In particular this applies to algebraic number fields, so any element formula_28 of an algebraic number field formula_1 can be written as a zero of a polynomial with rational coefficients. Therefore, elements of formula_1 are also referred to as "algebraic numbers". Given a polynomial formula_38 such that formula_39, it can be arranged such that the leading coefficient formula_40 is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients. If, however, its coefficients are actually all integers, formula_28 is called an "algebraic integer". Any (usual) integer formula_42 is an algebraic integer, as it is the zero of the linear monic polynomial:
It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in formula_1 form a ring denoted formula_45 called the ring of integers of It is a subring of (that is, a ring contained in) A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of formula_1 is an integral domain. The field formula_1 is the field of fractions of the integral domain This way one can get back and forth between the algebraic number field formula_1 and its ring of integers Rings of algebraic integers have three distinctive properties: firstly, formula_45 is an integral domain that is integrally closed in its field of fractions Secondly, formula_45 is a Noetherian ring. Finally, every nonzero prime ideal of formula_45 is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a "Dedekind ring" (or "Dedekind domain"), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
Unique factorization.
For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal formula_52 in the ring formula_53 of quadratic integers factors into prime ideals as
However, unlike formula_55 as the ring of integers of the ring of integers of a proper extension of formula_56 need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in the uniqueness of the factorization fails:
Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in Euclidean domains are unique factorization domains; for example the ring of Gaussian integers, and the ring of Eisenstein integers, where formula_58 is a cube root of unity (unequal to 1), have this property.
ζ-functions, "L"-functions and class number formula.
The failure of unique factorization is measured by the class number, commonly denoted "h", the cardinality of the so-called ideal class group. This group is always finite. The ring of integers formula_45 possesses unique factorization if and only if it is a principal ring or, equivalently, if formula_1 has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields (i.e., formula_61) with prescribed class number. The class number formula relates "h" to other fundamental invariants of It involves the Dedekind zeta function ζformula_1(s), a function in a complex variable "s", defined by
(The product is over all prime ideals of formula_64 denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field The infinite product converges only for Re("s") > 1, in general analytic continuation and the functional equation for the zeta-function are needed to define the function for all "s").
The Dedekind zeta-function generalizes the Riemann zeta-function in that ζformula_4("s") = ζ("s").
The class number formula states that ζformula_1("s") has a simple pole at "s" = 1 and at this point the residue is given by
Here "r"1 and "r"2 classically denote the number of real embeddings and pairs of complex embeddings of respectively. Moreover, Reg is the regulator of "w" the number of roots of unity in formula_1 and "D" is the discriminant of
Dirichlet L-functions formula_69 are a more refined variant of formula_70. Both types of functions encode the arithmetic behavior of formula_4 and formula_1, respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression
with coprime formula_74 and formula_75, there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet formula_76-function is nonzero at formula_77. Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.
Bases for number fields.
Integral basis.
An "integral basis" for a number field formula_1 of degree formula_17 is a set
of "n" algebraic integers in formula_1 such that every element of the ring of integers formula_45 of formula_1 can be written uniquely as a Z-linear combination of elements of "B"; that is, for any "x" in formula_45 we have
where the "mi" are (ordinary) integers. It is then also the case that any element of formula_1 can be written uniquely as
where now the "mi" are rational numbers. The algebraic integers of formula_1 are then precisely those elements of formula_1 where the "mi" are all integers.
Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.
Power basis.
Let formula_1 be a number field of degree Among all possible bases of formula_1 (seen as a formula_4-vector space), there are particular ones known as power bases, that are bases of the form
for some element By the primitive element theorem, there exists such an formula_32, called a primitive element. If formula_32 can be chosen in formula_45 and such that formula_94 is a basis of formula_45 as a free Z-module, then formula_94 is called a power integral basis, and the field formula_1 is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial
formula_98
Regular representation, trace and discriminant.
Recall that any field extension formula_99 has a unique formula_4-vector space structure. Using the multiplication in formula_1, an element formula_32 of the field formula_1 over the base field formula_4 may be represented by formula_105 matrices
by requiring
Here formula_108 is a fixed basis for formula_1, viewed as a formula_4-vector space. The rational numbers formula_111 are uniquely determined by formula_32 and the choice of a basis since any element of formula_1 can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field formula_1 is called the "regular representation". The square matrix formula_115 represents the effect of multiplication by formula_32 in the given basis. It follows that if the element formula_117 of formula_1 is represented by a matrix formula_119, then the product formula_120 is represented by the matrix product formula_121. Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element formula_32 and not on the basis. In particular, the trace of the matrix formula_123 is called the "trace" of the field element formula_32 and denoted formula_125, and the determinant is called the "norm" of "x" and denoted formula_126.
Now this can be generalized slightly by instead considering a field extension formula_127 and giving an formula_76-basis for formula_1. Then, there is an associated matrix formula_130 which has trace formula_131 and norm formula_132 defined as the trace and determinant of the matrix formula_130.
Example.
Consider the field extension formula_134 where formula_135. Then, we have a formula_4-basis given byformula_137since any formula_138 can be expressed as some formula_4-linear combinationformula_140Then, we can take some formula_141 where formula_142 and compute formula_143. Writing this out givesformula_144We can find the matrix formula_123 by writing out the associated matrix equation giving formula_146showingformula_147We can then compute the trace and determinant with relative ease, giving the trace and norm.
Properties.
By definition, standard properties of traces and determinants of matrices carry over to Tr and N: Tr("x") is a linear function of "x", as expressed by , , and the norm is a multiplicative homogeneous function of degree "n": , . Here "λ" is a rational number, and "x", "y" are any two elements of
The "trace form" derived is a bilinear form defined by means of the trace, asformula_148 by formula_149formula_131. The integral trace form, an integer-valued symmetric matrix is defined as formula_151, where "b"1, …, "b"n is an integral basis for The "discriminant" of formula_1 is defined as det("t"). It is an integer, and is an invariant property of the field formula_1, not depending on the choice of integral basis.
The matrix associated to an element "x" of formula_1 can also be used to give other, equivalent descriptions of algebraic integers. An element "x" of formula_1 is an algebraic integer if and only if the characteristic polynomial "p""A" of the matrix "A" associated to "x" is a monic polynomial with integer coefficients. Suppose that the matrix "A" that represents an element "x" has integer entries in some basis "e". By the Cayley–Hamilton theorem, "p""A"("A") = 0, and it follows that "p""A"("x") = 0, so that "x" is an algebraic integer. Conversely, if "x" is an element of formula_1 which is a root of a monic polynomial with integer coefficients then the same property holds for the corresponding matrix "A". In this case it can be proven that "A" is an integer matrix in a suitable basis of The property of being an algebraic integer is "defined" in a way that is independent of a choice of a basis in
Example with integral basis.
Consider formula_157, where "x" satisfies . Then an integral basis is [1, "x", 1/2("x"2 + 1)], and the corresponding integral trace form is
The "3" in the upper left hand corner of this matrix is the trace of the matrix of the map defined by the first basis element (1) in the regular representation of formula_1 on This basis element induces the identity map on the 3-dimensional vector space, The trace of the matrix of the identity map on a 3-dimensional vector space is 3.
The determinant of this is , the field discriminant; in comparison the root discriminant, or discriminant of the polynomial, is .
Places.
Mathematicians of the nineteenth century assumed that algebraic numbers were a type of complex number. This situation changed with the discovery of p-adic numbers by Hensel in 1897; and now it is standard to consider all of the various possible embeddings of a number field formula_1 into its various topological completions formula_161 at once.
A "place" of a number field formula_1 is an equivalence class of absolute values on formula_1pg 9. Essentially, an absolute value is a notion to measure the size of elements formula_32 of Two such absolute values are considered equivalent if they give rise to the same notion of smallness (or proximity). The equivalence relation between absolute values formula_165 is given by some formula_166 such thatformula_167meaning we take the value of the norm formula_168 to the formula_169-th power.
For the following non-trivial norms occur (Ostrowski's theorem): the (usual) absolute value, sometimes denoted formula_178 which gives rise to the complete topological field of the real numbers On the other hand, for any prime number formula_38, the "p"-adic absolute values is defined by
are used to construct the formula_38-adic numbers In contrast to the usual absolute value, the "p"-adic norm gets "smaller" when "q" is multiplied by "p", leading to quite different behavior of formula_181 vis-à-vis
Note the general situation typically considered is taking a number field formula_1 and considering a prime ideal formula_183 for its associated ring of algebraic numbers Then, there will be a unique place formula_184 called a non-archimedian place. In addition, for every embedding formula_185 there will be a place called an archimedian place, denoted This statement is a theorem also called Ostrowski's theorem.
Examples.
The field formula_186 for formula_187 where formula_188 is a fixed 6th root of unity, provides a rich example for constructing explicit real and complex archimedian embeddings, and non-archimedian embeddings as wellpg 15-16.
Archimedean places.
Here we use the standard notation formula_189 and formula_190 for the number of real and complex embeddings used, respectively (see below).
Calculating the archimedean places of a number field formula_1 is done as follows: let formula_32 be a primitive element of formula_1, with minimal polynomial formula_28 (over formula_4). Over formula_23, formula_28 will generally no longer be irreducible, but its irreducible (real) factors are either of degree one or two. Since there are no repeated roots, there are no repeated factors. The roots formula_198 of factors of degree one are necessarily real, and replacing formula_32 by formula_198 gives an embedding of formula_1 into formula_23; the number of such embeddings is equal to the number of real roots of Restricting the standard absolute value on formula_23 to formula_1 gives an archimedean absolute value on formula_1; such an absolute value is also referred to as a real place of On the other hand, the roots of factors of degree two are pairs of conjugate complex numbers, which allows for two conjugate embeddings into Either one of this pair of embeddings can be used to define an absolute value on formula_1, which is the same for both embeddings since they are conjugate. This absolute value is called a complex place of
If all roots of formula_28 above are real (respectively, complex) or, equivalently, any possible embedding formula_208 is actually forced to be inside formula_23 (resp. formula_1 is called totally real (resp. totally complex).
Non-Archimedean or ultrametric places.
To find the non-Archimedean places, let again formula_28 and formula_32 be as above. In formula_28 splits in factors of various degrees, none of which are repeated, and the degrees of which add up to the degree of For each of these formula_38-adically irreducible factors we may suppose that formula_32 satisfies formula_216 and obtain an embedding of formula_1 into an algebraic extension of finite degree over Such a local field behaves in many ways like a number field, and the formula_38-adic numbers may similarly play the role of the rationals; in particular, we can define the norm and trace in exactly the same way, now giving functions mapping to By using this formula_38-adic norm map formula_220 for the place formula_216, we may define an absolute value corresponding to a given formula_38-adically irreducible factor formula_216 of degree formula_75 byformula_225Such an absolute value is called an ultrametric, non-Archimedean or formula_38-adic place of
For any ultrametric place "v" we have that |"x"|"v" ≤ 1 for any "x" in since the minimal polynomial for "x" has integer factors, and hence its "p"-adic factorization has factors in Z"p". Consequently, the norm term (constant term) for each factor is a "p"-adic integer, and one of these is the integer used for defining the absolute value for "v".
Prime ideals in "OK".
For an ultrametric place "v", the subset of formula_45 defined by |"x"|"v" < 1 is an ideal formula_228 of This relies on the ultrametricity of "v": given "x" and "y" in then
Actually, formula_228 is even a prime ideal.
Conversely, given a prime ideal formula_228 of a discrete valuation can be defined by setting formula_231 where "n" is the biggest integer such that the "n"-fold power of the ideal. This valuation can be turned into an ultrametric place. Under this correspondence, (equivalence classes) of ultrametric places of formula_1 correspond to prime ideals of For this gives back Ostrowski's theorem: any prime ideal in Z (which is necessarily by a single prime number) corresponds to a non-Archimedean place and vice versa. However, for more general number fields, the situation becomes more involved, as will be explained below.
Yet another, equivalent way of describing ultrametric places is by means of localizations of Given an ultrametric place formula_233 on a number field the corresponding localization is the subring formula_234 of formula_1 of all elements formula_32 such that | "x" |"v" ≤ 1. By the ultrametric property formula_234 is a ring. Moreover, it contains For every element "x" of at least one of "x" or "x"−1 is contained in Actually, since "K"×/"T"× can be shown to be isomorphic to the integers, formula_234 is a discrete valuation ring, in particular a local ring. Actually, formula_234 is just the localization of formula_45 at the prime ideal so </math>.}} Conversely, formula_228 is the maximal ideal of
Altogether, there is a three-way equivalence between ultrametric absolute values, prime ideals, and localizations on a number field.
Lying over theorem and places.
One of the basic theorems in algebraic number theory are the going up and going down theorems, which describe the behavior of some prime ideal formula_183 when it is extended as an ideal in formula_243 for some field extension We say that an ideal formula_244 lies over formula_228 if Then, one incarnation of the theorem states a prime ideal in formula_246 lies over hence there is always a surjective mapformula_247induced from the inclusion Since there exists a correspondence between places and prime ideals, this means we can find places dividing a place which is induced from a field extension. That is, if formula_38 is a place of then there are places formula_233 of formula_76 which divide in the sense that their induced prime ideals divide the induced prime ideal of formula_38 in
In fact, this observation is usefulpg 13 while looking at the base change of an algebraic field extension of formula_4 to one of its completions If we writeformula_253and write formula_254 for the induced element of we get a decomposition of Explicitly, this decomposition isformula_255furthermore, the induced polynomial formula_256 decomposes asformula_257because of Hensel's lemmapg 129-131, henceformula_258Moreover, there are embeddingsformula_259where formula_260 is a root of formula_261 giving formula_262, hence we could writeformula_263as subsets of formula_264 (which is the completion of the algebraic closure of
Ramification.
Ramification, generally speaking, describes a geometric phenomenon that can occur with finite-to-one maps (that is, maps formula_265 such that the preimages of all points "y" in "Y" consist only of finitely many points): the cardinality of the fibers "f"−1("y") will generally have the same number of points, but it occurs that, in special points "y", this number drops. For example, the map
has "n" points in each fiber over "t", namely the "n" (complex) roots of "t", except in t = "0", where the fiber consists of only one element, "z" = 0. One says that the map is "ramified" in zero. This is an example of a branched covering of Riemann surfaces. This intuition also serves to define ramification in algebraic number theory. Given a (necessarily finite) extension of number fields formula_127, a prime ideal "p" of formula_243 generates the ideal "pO""K" of This ideal may or may not be a prime ideal, but, according to the Lasker–Noether theorem (see above), always is given by
with uniquely determined prime ideals "q""i" of formula_45 and numbers (called ramification indices) "e""i". Whenever one ramification index is bigger than one, the prime "p" is said to ramify in
The connection between this definition and the geometric situation is delivered by the map of spectra of rings In fact, unramified morphisms of schemes in algebraic geometry are a direct generalization of unramified extensions of number fields.
Ramification is a purely local property, i.e., depends only on the completions around the primes "p" and "q""i". The inertia group measures the difference between the local Galois groups at some place and the Galois groups of the involved finite residue fields.
An example.
The following example illustrates the notions introduced above. In order to compute the ramification index of where
at 23, it suffices to consider the field extension Up to 529 = 232 (i.e., modulo 529) "f" can be factored as
Substituting in the first factor "g" modulo 529 yields "y" + 191, so the valuation | "y" |"g" for "y" given by "g" is | −191 |23 = 1. On the other hand, the same substitution in "h" yields Since 161 = 7 × 23,
Since possible values for the absolute value of the place defined by the factor "h" are not confined to integer powers of 23, but instead are integer powers of the square root of 23, the ramification index of the field extension at 23 is two.
The valuations of any element of formula_1 can be computed in this way using resultants. If, for example "y" = "x"2 − "x" − 1, using the resultant to eliminate "x" between this relationship and "f" = "x"3 − "x" − 1 = 0 gives . If instead we eliminate with respect to the factors "g" and "h" of "f", we obtain the corresponding factors for the polynomial for "y", and then the 23-adic valuation applied to the constant (norm) term allows us to compute the valuations of "y" for "g" and "h" (which are both 1 in this instance.)
Dedekind discriminant theorem.
Much of the significance of the discriminant lies in the fact that ramified ultrametric places are all places obtained from factorizations in formula_181 where "p" divides the discriminant. This is even true of the polynomial discriminant; however the converse is also true, that if a prime "p" divides the discriminant, then there is a "p"-place which ramifies. For this converse the field discriminant is needed. This is the Dedekind discriminant theorem. In the example above, the discriminant of the number field formula_274 with "x"3 − "x" − 1 = 0 is −23, and as we have seen the 23-adic place ramifies. The Dedekind discriminant tells us it is the only ultrametric place which does. The other ramified place comes from the absolute value on the complex embedding of formula_1.
Galois groups and Galois cohomology.
Generally in abstract algebra, field extensions "K" / "L" can be studied by examining the Galois group Gal("K" / "L"), consisting of field automorphisms of formula_1 leaving formula_76 elementwise fixed. As an example, the Galois group formula_278 of the cyclotomic field extension of degree "n" (see above) is given by (Z/"nZ)×, the group of invertible elements in Z/"nZ. This is the first stepstone into Iwasawa theory.
In order to include all possible extensions having certain properties, the Galois group concept is commonly applied to the (infinite) field extension / "K" of the algebraic closure, leading to the absolute Galois group "G" := Gal( / "K") or just Gal("K"), and to the extension formula_2. The fundamental theorem of Galois theory links fields in between formula_1 and its algebraic closure and closed subgroups of Gal("K"). For example, the abelianization (the biggest abelian quotient) "G"ab of "G" corresponds to a field referred to as the maximal abelian extension "K"ab (called so since any further extension is not abelian, i.e., does not have an abelian Galois group). By the Kronecker–Weber theorem, the maximal abelian extension of formula_4 is the extension generated by all roots of unity. For more general number fields, class field theory, specifically the Artin reciprocity law gives an answer by describing "G"ab in terms of the idele class group. Also notable is the Hilbert class field, the maximal abelian unramified field extension of formula_1. It can be shown to be finite over formula_1, its Galois group over formula_1 is isomorphic to the class group of formula_1, in particular its degree equals the class number "h" of formula_1 (see above).
In certain situations, the Galois group acts on other mathematical objects, for example a group. Such a group is then also referred to as a Galois module. This enables the use of group cohomology for the Galois group Gal("K"), also known as Galois cohomology, which in the first place measures the failure of exactness of taking Gal("K")-invariants, but offers deeper insights (and questions) as well. For example, the Galois group "G" of a field extension "L" / "K" acts on "L"×, the nonzero elements of "L". This Galois module plays a significant role in many arithmetic dualities, such as Poitou-Tate duality. The Brauer group of originally conceived to classify division algebras over formula_1, can be recast as a cohomology group, namely H2(Gal ("K", ×).
Local-global principle.
Generally speaking, the term "local to global" refers to the idea that a global problem is first done at a local level, which tends to simplify the questions. Then, of course, the information gained in the local analysis has to be put together to get back to some global statement. For example, the notion of sheaves reifies that idea in topology and geometry.
Local and global fields.
Number fields share a great deal of similarity with another class of fields much used in algebraic geometry known as function fields of algebraic curves over finite fields. An example is "K""p"("T"). They are similar in many respects, for example in that number rings are one-dimensional regular rings, as are the coordinate rings (the quotient fields of which is the function field in question) of curves. Therefore, both types of field are called global fields. In accordance with the philosophy laid out above, they can be studied at a local level first, that is to say, by looking at the corresponding local fields. For number fields the local fields are the completions of formula_1 at all places, including the archimedean ones (see local analysis). For function fields, the local fields are completions of the local rings at all points of the curve for function fields.
Many results valid for function fields also hold, at least if reformulated properly, for number fields. However, the study of number fields often poses difficulties and phenomena not encountered in function fields. For example, in function fields, there is no dichotomy into non-archimedean and archimedean places. Nonetheless, function fields often serves as a source of intuition what should be expected in the number field case.
Hasse principle.
A prototypical question, posed at a global level, is whether some polynomial equation has a solution in If this is the case, this solution is also a solution in all completions. The local-global principle or Hasse principle asserts that for quadratic equations, the converse holds, as well. Thereby, checking whether such an equation has a solution can be done on all the completions of which is often easier, since analytic methods (classical analytic tools such as intermediate value theorem at the archimedean places and p-adic analysis at the nonarchimedean places) can be used. This implication does not hold, however, for more general types of equations. However, the idea of passing from local data to global ones proves fruitful in class field theory, for example, where local class field theory is used to obtain global insights mentioned above. This is also related to the fact that the Galois groups of the completions "K"v can be explicitly determined, whereas the Galois groups of global fields, even of formula_4 are far less understood.
Adeles and ideles.
In order to assemble local data pertaining to all local fields attached to the adele ring is set up. A multiplicative variant is referred to as ideles.


Group of symplectic type

In mathematical finite group theory, a p-group of symplectic type is a "p"-group such that all characteristic abelian subgroups are cyclic.
According to , the "p"-groups of symplectic type were classified by P. Hall in unpublished lecture notes, who showed that they are all a central product of an extraspecial group with a group that is cyclic, dihedral, quasidihedral, or quaternion. gives a proof of this result.
The width "n" of a group "G" of symplectic type is the largest integer "n" such that the group contains an extraspecial subgroup "H" of order "p"1+2"n" such that "G" = "H"."C""G"("H"), or 0 if "G" contains no such subgroup.
Groups of symplectic type appear in centralizers of involutions of groups of GF(2)-type.


Semi-abelian category

In mathematics, specifically in category theory, a semi-abelian category is a pre-abelian category in which the induced morphism formula_1 is a bimorphism, i.e., a monomorphism and an epimorphism, for every morphism formula_2.
Properties.
The two properties used in the definition can be characterized by several equivalent conditions.
Every semi-abelian category has a maximal exact structure.
If a semi-abelian category is not quasi-abelian, then the class of all kernel-cokernel pairs does not form an exact structure.
Examples.
Every quasi-abelian category is semi-abelian. In particular, every abelian category is semi-abelian. Non quasi-abelian examples are the following.
and formula_4 be a field. The category of finitely generated projective modules over the algebra formula_5 is semi-abelian.
History.
The concept of a semi-abelian category was developed in the 1960s. Raikov conjectured that the notion of a quasi-abelian category is equivalent to that of a semi-abelian category. Around 2005 it turned out that the conjecture is false.
Left and right semi-abelian categories.
By dividing the two conditions on the induced map in the definition, one can define left semi-abelian categories by requiring that formula_6 is a monomorphism for each morphism formula_2. Accordingly, right semi-abelian categories are pre-abelian categories such that formula_6 is an epimorphism for each morphism formula_2.
If a category is left semi-abelian and right quasi-abelian, then it is already quasi-abelian. The same holds, if the category is right semi-abelian and left quasi-abelian.


Vogel plane

In mathematics, the Vogel plane is a method of parameterizing simple Lie algebras by eigenvalues α, β, γ of the Casimir operator on the symmetric square of the Lie algebra, which gives a point (α: β: γ) of "P"2/"S"3, the projective plane "P"2 divided out by the symmetric group "S"3 of permutations of coordinates. It was introduced by , and is related by some observations made by . generalized Vogel's work to higher symmetric powers.
The point of the projective plane (modulo permutations) corresponding to a simple complex Lie algebra is given by three eigenvalues α, β, γ of the Casimir operator acting on spaces "A", "B", "C", where the symmetric square of the Lie algebra (usually) decomposes as a sum of the complex numbers and 3 irreducible spaces "A", "B", "C".


Unreferenced variable

An unreferenced variable in the source code of a computer program is a variable that is defined but which is never used. This may result in a harmless waste of memory. Many compilers detect such variables and do not allocate storage for them (i.e., "optimize away" their storage), generally also issuing a warning as they do.
Some coding guideline documents consider an unreferenced variable to be a symptom of a potential coding fault. On the other hand, unreferenced variables can be used as temporary placeholders to indicate further expected future developments in the code.
Examples.
C:
 int main(void)
 int i, j;
 for (i=0; i<10; i++)
 printf("%d", i);
 return 0;
In this example, j is an unreferenced variable.


Forgetful functor

In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given signature, this may be expressed by curtailing the signature: the new signature is an edited form of the old one. If the signature is left as an empty list, the functor is simply to take the underlying set of a structure. Because many structures in mathematics consist of a set with an additional added structure, a forgetful functor that maps to the underlying set is the most common case.
Overview.
As an example, there are several forgetful functors from the category of commutative rings. A (unital) ring, described in the language of universal algebra, is an ordered tuple ("R", +, ×, "a", 0, 1) satisfying certain axioms, where "+" and "×" are binary functions on the set "R", "a" is a unary operation corresponding to additive inverse, and 0 and 1 are nullary operations giving the identities of the two binary operations. Deleting the 1 gives a forgetful functor to the category of rings without unit; it simply "forgets" the unit. Deleting "×" and 1 yields a functor to the category of abelian groups, which assigns to each ring "R" the underlying additive abelian group of "R". To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups. Deleting all the operations gives the functor to the underlying set "R".
It is beneficial to distinguish between forgetful functors that "forget structure" versus those that "forget properties". For example, in the above example of commutative rings, in addition to those functors that delete some of the operations, there are functors that forget some of the axioms. There is a functor from the category CRing to Ring that forgets the axiom of commutativity, but keeps all the operations. Occasionally the object may include extra sets not defined strictly in terms of the underlying set (in this case, which part to consider the underlying set is a matter of taste, though this is rarely ambiguous in practice). For these objects, there are forgetful functors that forget the extra sets that are more general.
Most common objects studied in mathematics are constructed as underlying sets along with extra sets of structure on those sets (operations on the underlying set, privileged subsets of the underlying set, etc.) which may satisfy some axioms. For these objects, a commonly considered forgetful functor is as follows.
Let formula_1 be any category based on sets, e.g. groups—sets of elements—or topological spaces—sets of 'points'. As usual, write formula_2 for the objects of formula_1 and write formula_4 for the morphisms of the same. Consider the rule:
The functor formula_11 is then the forgetful functor from formula_1 to Set, the category of sets.
Forgetful functors are almost always faithful. Concrete categories have forgetful functors to the category of sets—indeed they may be "defined" as those categories that admit a faithful functor to that category.
Forgetful functors that only forget axioms are always fully faithful, since every morphism that respects the structure between objects that satisfy the axioms automatically also respects the axioms. Forgetful functors that forget structures need not be full; some morphisms don't respect the structure. These functors are still faithful however because distinct morphisms that do respect the structure are still distinct when the structure is forgotten. Functors that forget the extra sets need not be faithful, since distinct morphisms respecting the structure of those extra sets may be indistinguishable on the underlying set.
In the language of formal logic, a functor of the first kind removes axioms, a functor of the second kind removes predicates, and a functor of the third kind remove types. An example of the first kind is the forgetful functor Ab → Grp. One of the second kind is the forgetful functor Ab → Set. A functor of the third kind is the functor Mod → Ab, where Mod is the fibred category of all modules over arbitrary rings. To see this, just choose a ring homomorphism between the underlying rings that does not change the ring action. Under the forgetful functor, this morphism yields the identity. Note that an object in Mod is a tuple, which includes a ring and an abelian group, so which to forget is a matter of taste.
Left adjoints of forgetful functors.
Forgetful functors tend to have left adjoints, which are 'free' constructions. For example:
For a more extensive list, see (Mac Lane 1997).
As this is a fundamental example of adjoints, we spell it out:
adjointness means that given a set "X" and an object (say, an "R"-module) "M", maps "of sets" formula_20 correspond to maps of modules formula_21: every map of sets yields a map of modules, and every map of modules comes from a map of sets.
In the case of vector spaces, this is summarized as:
"A map between vector spaces is determined by where it sends a basis, and a basis can be mapped to anything."
Symbolically:
The unit of the free–forgetful adjunction is the "inclusion of a basis": formula_23.
Fld, the category of fields, furnishes an example of a forgetful functor with no adjoint. There is no field satisfying a free universal property for a given set.


Tropical compactification

In algebraic geometry, a tropical compactification is a compactification (projective completion) of a subvariety of an algebraic torus, introduced by Jenia Tevelev. Given an algebraic torus and a connected closed subvariety of that torus, a compatification of the subvariety is defined as a closure of it in a toric variety of the original torus. The concept of a tropical compatification arises when trying to make compactifications as "nice" as possible. For a torus formula_1, a toric variety formula_2, the compatification formula_3 is tropical when the map
is faithfully flat and formula_3 is proper.


Generalized dihedral group

In mathematics, the generalized dihedral groups are a family of groups with algebraic structures similar to that of the dihedral groups. They include the finite dihedral groups, the infinite dihedral group, and the orthogonal group "O"(2). Dihedral groups play an important role in group theory, geometry, and chemistry.
Definition.
For any abelian group "H", the generalized dihedral group of "H", written Dih("H"), is the semidirect product of "H" and Z2, with Z2 acting on "H" by inverting elements. I.e., formula_1 with φ(0) the identity and φ(1) inversion.
Thus we get:
for all "h"1, "h"2 in "H" and "t"2 in Z2.
Note that ("h", 0) * (0,1) = ("h",1), i.e. first the inversion and then the operation in "H". Also (0, 1) * ("h", "t") = (−"h", 1 + "t"); indeed (0,1) inverts "h", and toggles "t" between "normal" (0) and "inverted" (1) (this combined operation is its own inverse).
The subgroup of Dih("H") of elements ("h", 0) is a normal subgroup of index 2, isomorphic to "H", while the elements ("h", 1) are all their own inverse.
The conjugacy classes are:
Thus for every subgroup "M" of "H", the corresponding set of elements ("m",0) is also a normal subgroup. We have:
Properties.
Dih("H") is Abelian, with the semidirect product a direct product, if and only if all elements of "H" are their own inverse, i.e., an elementary abelian 2-group:
etc.
Topology.
Dih(R"n" ) and its dihedral subgroups are disconnected topological groups. Dih(R"n" ) consists of two connected components: the identity component isomorphic to R"n", and the component with the reflections. Similarly O(2) consists of two connected components: the identity component isomorphic to the circle group, and the component with the reflections.
For the group Dih∞ we can distinguish two cases:
Both topological groups are totally disconnected, but in the first case the (singleton) components are open, while in the second case they are not. Also, the first topological group is a closed subgroup of Dih(R) but the second is not a closed subgroup of O(2).


Positive polynomial

In mathematics, a positive polynomial on a particular set is a polynomial whose values are positive on that set.
Let "p" be a polynomial in "n" variables with real coefficients and let "S" be a subset of the "n"-dimensional Euclidean space ℝ"n". We say that:
For certain sets "S", there exist algebraic descriptions of all polynomials that are positive, non-negative, or zero on "S". Such a description is a positivstellensatz, nichtnegativstellensatz, or nullstellensatz. This article will focus on the former two descriptions. For the latter, see Hilbert's Nullstellensatz for the most known nullstellensatz.
Generalizations of positivstellensatz.
Positivstellensatz also exist for trigonometric polynomials, matrix polynomials, polynomials in free variables, various quantum polynomials, etc.


Explained sum of squares

In statistics, the explained sum of squares (ESS), alternatively known as the model sum of squares or sum of squares due to regression ("SSR" – not to be confused with the residual sum of squares RSS or sum of squares of errors), is a quantity used in describing how well a model, often a regression model, represents the data being modelled. In particular, the explained sum of squares measures how much variation there is in the modelled values and this is compared to the total sum of squares ( TSS ), which measures how much variation there is in the observed data, and to the residual sum of squares, which measures the variation in the error between the observed data and modelled values.
Definition.
The explained sum of squares (ESS) is the sum of the squares of the deviations of the predicted values from the mean value of a response variable, in a standard regression model — for example, , where "y""i" is the "i" th observation of the response variable, "x""ji" is the "i" th observation of the "j" th explanatory variable, "a" and "b""j" are coefficients, "i" indexes the observations from 1 to "n", and "ε""i" is the "i" th value of the error term. In general, the greater the ESS, the better the estimated model performs.
If formula_1 and formula_2 are the estimated coefficients, then
is the "i" th predicted value of the response variable. The ESS is then:
In some cases (see below): total sum of squares ( TSS ) = explained sum of squares (ESS)+ residual sum of squares (RSS).
Partitioning in simple linear regression.
The following equality, stating that the total sum of squares (TSS) equals the residual sum of squares (=SSE : the sum of squared errors of prediction) plus the explained sum of squares (SSR :the sum of squares due to regression or explained sum of squares), is generally true in simple linear regression:
Simple derivation.
Square both sides and sum over all "i":
Here is how the last term above is zero from simple linear regression
So,
Therefore,
Partitioning in the general ordinary least squares model.
The general regression model with "n" observations and "k" explanators, the first of which is a constant unit vector whose coefficient is the regression intercept, is
where "y" is an "n" × 1 vector of dependent variable observations, each column of the "n" × "k" matrix "X" is a vector of observations on one of the "k" explanators, formula_16 is a "k" × 1 vector of true coefficients, and "e" is an "n" × 1 vector of the true underlying errors. The ordinary least squares estimator for formula_17 is
The residual vector formula_19 is formula_20, so the residual sum of squares formula_21 is, after simplification,
Denote as formula_23 the constant vector all of whose elements are the sample mean formula_24 of the dependent variable values in the vector "y". Then the total sum of squares is
The explained sum of squares, defined as the sum of squared deviations of the predicted values from the observed mean of "y", is
Using formula_27 in this, and simplifying to obtain formula_28, gives the result that "TSS" = "ESS" + "RSS" if and only if formula_29. The left side of this is formula_24 times the sum of the elements of "y", and the right side is formula_24 times the sum of the elements of formula_32, so the condition is that the sum of the elements of "y" equals the sum of the elements of formula_32, or equivalently that the sum of the prediction errors (residuals) formula_34 is zero. This can be seen to be true by noting the well-known OLS property that the "k" × 1 vector formula_35: since the first column of "X" is a vector of ones, the first element of this vector formula_36 is the sum of the residuals and is equal to zero. This proves that the condition holds for the result that "TSS" = "ESS" + "RSS".
In linear algebra terms, we have formula_37, formula_38, formula_39.
The proof can be simplified by noting that formula_40. The proof is as follows:
Thus,
which again gives the result that "TSS" = "ESS" + "RSS", since formula_43.


Picard modular group

In mathematics, a Picard modular group, studied by , is a group of the form SU("J","L"), where "L" is a 3-dimensional lattice over the ring of integers of an imaginary quadratic field and "J" is a hermitian form on "L" of signature (2, 1). Picard modular groups act on the unit sphere in C2 and the quotient is called a Picard modular surface.


Pullback (category theory)

In category theory, a branch of mathematics, a pullback (also called a fiber product, fibre product, fibered product or Cartesian square) is the limit of a diagram consisting of two morphisms and with a common codomain. The pullback is often written
and comes equipped with two natural morphisms and . The pullback of two morphisms and need not exist, but if it does, it is essentially uniquely defined by the two morphisms. In many situations, may intuitively be thought of as consisting of pairs of elements with in , in , and . For the general definition, a universal property is used, which essentially expresses the fact that the pullback is the "most general" way to complete the two given morphisms to a commutative square.
The dual concept of the pullback is the "pushout".
Universal property.
Explicitly, a pullback of the morphisms and consists of an object and two morphisms and for which the diagram
commutes. Moreover, the pullback must be universal with respect to this diagram. That is, for any other such triple where and are morphisms with , there must exist a unique such that
This situation is illustrated in the following commutative diagram.
As with all universal constructions, a pullback, if it exists, is unique up to isomorphism. In fact, given two pullbacks and of the same cospan , there is a unique isomorphism between and respecting the pullback structure.
Pullback and product.
The pullback is similar to the product, but not the same. One may obtain the product by "forgetting" that the morphisms and exist, and forgetting that the object exists. One is then left with a discrete category containing only the two objects and , and no arrows between them. This discrete category may be used as the index set to construct the ordinary binary product. Thus, the pullback can be thought of as the ordinary (Cartesian) product, but with additional structure. Instead of "forgetting" , , and , one can also "trivialize" them by specializing to be the terminal object (assuming it exists). and are then uniquely determined and thus carry no information, and the pullback of this cospan can be seen to be the product of and .
Examples.
Commutative rings.
In the category of commutative rings (with identity), the pullback is called the fibered product. Let , , and be commutative rings (with identity) and and (identity preserving) ring homomorphisms. Then the pullback of this diagram exists and given by the subring of the product ring defined by
along with the morphisms
given by formula_4 and formula_5 for all formula_6. We then have
Groups, Modules.
In complete analogy to the example of commutative rings above, one can show that all pullbacks exist in the category of groups and in the category of modules over some fixed ring.
Sets.
In the category of sets, the pullback of functions and always exists and is given by the set
together with the restrictions of the projection maps and to .
Alternatively one may view the pullback in asymmetrically:
where formula_10 is the disjoint union of sets (the involved sets are not disjoint on their own unless resp. is injective). In the first case, the projection extracts the index while forgets the index, leaving elements of .
This example motivates another way of characterizing the pullback: as the equalizer of the morphisms where is the binary product of and and and are the natural projections. This shows that pullbacks exist in any category with binary products and equalizers. In fact, by the existence theorem for limits, all finite limits exist in a category with binary products and equalizers; equivalently, all finite limits exist in a category with terminal object and pullbacks (by the fact that binary product = pullback on the terminal object, and that an equalizer is a pullback involving binary product).
Fiber bundles.
Another example of a pullback comes from the theory of fiber bundles: given a bundle map and a continuous map , the pullback (formed in the category of topological spaces with continuous maps) is a fiber bundle over called the pullback bundle. The associated commutative diagram is a morphism of fiber bundles.
Preimages and intersections.
Preimages of sets under functions can be described as pullbacks as follows:
Suppose , . Let be the inclusion map . Then a pullback of and (in ) is given by the preimage together with the inclusion of the preimage in
and the restriction of to
Because of this example, in a general category the pullback of a morphism and a monomorphism can be thought of as the "preimage" under of the subobject specified by . Similarly, pullbacks of two monomorphisms can be thought of as the "intersection" of the two subobjects.
Least common multiple.
Consider the multiplicative monoid of positive integers as a category with one object. In this category, the pullback of two positive integers and is just the pair , where the numerators are both the least common multiple of and . The same pair is also the pushout.
Properties.
formula_11
formula_12
Weak pullbacks.
A weak pullback of a cospan is a cone over the cospan that is only weakly universal, that is, the mediating morphism above is not required to be unique.


Vector addition


Subnormal subgroup

In mathematics, in the field of group theory, a subgroup "H" of a given group "G" is a subnormal subgroup of "G" if there is a finite chain of subgroups of the group, each one normal in the next, beginning at "H" and ending at "G".
In notation, formula_1 is formula_2-subnormal in formula_3 if there are subgroups
of formula_3 such that formula_6 is normal in formula_7 for each formula_8.
A subnormal subgroup is a subgroup that is formula_2-subnormal for some positive integer formula_2.
Some facts about subnormal subgroups:
The property of subnormality is transitive, that is, a subnormal subgroup of a subnormal
subgroup is subnormal. The relation of subnormality can be defined as the transitive closure of the relation of normality.
If every subnormal subgroup of "G" is normal in "G", then "G" is called a T-group.


Idempotent (ring theory)

In ring theory (part of abstract algebra) an idempotent element, or simply an idempotent, of a ring is an element "a" such that . That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that for any positive integer "n". For example, an idempotent element of a matrix ring is precisely an idempotent matrix.
For general rings, elements idempotent under multiplication are involved in decompositions of modules, and connected to homological properties of the ring. In Boolean algebra, the main objects of study are rings in which all elements are idempotent under both addition and multiplication.
Examples.
Quotients of Z.
One may consider the ring of integers mod "n", where "n" is squarefree. By the Chinese remainder theorem, this ring factors into the direct product of rings of integers mod "p". Now each of these factors is a field, so it is clear that the factor's only idempotents will be 0 and 1. That is, each factor has two idempotents. So if there are "m" factors, there will be 2"m" idempotents.
We can check this for the integers mod 6, . Since 6 has two factors (2 and 3) it should have 22 idempotents.
From these computations, 0, 1, 3, and 4 are idempotents of this ring, while 2 and 5 are not. This also demonstrates the decomposition properties described below: because , there is a ring decomposition . In 3Z/6Z the identity is 3+6Z and in 4Z/6Z the identity is 4+6Z.
Quotient of polynomial ring.
Given a ring formula_1 and an element formula_2 such that formula_3, then the quotient ring
formula_4
has the idempotent formula_5. For example, this could be applied to formula_6, or any polynomial formula_7.
Idempotents in split-quaternion rings.
There is a catenoid of idempotents in the split-quaternion ring.
Types of ring idempotents.
A partial list of important types of idempotents includes:
Any non-trivial idempotent "a" is a zero divisor (because with neither "a" nor "b" being zero, where ). This shows that integral domains and division rings do not have such idempotents. Local rings also do not have such idempotents, but for a different reason. The only idempotent contained in the Jacobson radical of a ring is 0.
Role in decompositions.
The idempotents of "R" have an important connection to decomposition of "R" modules. If "M" is an "R" module and is its ring of endomorphisms, then if and only if there is a unique idempotent "e" in "E" such that and . Clearly then, "M" is directly indecomposable if and only if 0 and 1 are the only idempotents in "E".
In the case when the endomorphism ring , where each endomorphism arises as left multiplication by a fixed ring element. With this modification of notation, as right modules if and only if there exists a unique idempotent "e" such that and . Thus every direct summand of "R" is generated by an idempotent.
If "a" is a central idempotent, then the corner ring is a ring with multiplicative identity "a". Just as idempotents determine the direct decompositions of "R" as a module, the central idempotents of "R" determine the decompositions of "R" as a direct sum of rings. If "R" is the direct sum of the rings "R"1...,"R""n", then the identity elements of the rings "R""i" are central idempotents in "R", pairwise orthogonal, and their sum is 1. Conversely, given central idempotents "a"1...,"a""n" in "R" that are pairwise orthogonal and have sum 1, then "R" is the direct sum of the rings "Ra"1,…,"Ra""n". So in particular, every central idempotent "a" in "R" gives rise to a decomposition of "R" as a direct sum of the corner rings "aRa" and . As a result, a ring "R" is directly indecomposable as a ring if and only if the identity 1 is centrally primitive.
Working inductively, one can attempt to decompose 1 into a sum of centrally primitive elements. If 1 is centrally primitive, we are done. If not, it is a sum of central orthogonal idempotents, which in turn are primitive or sums of more central idempotents, and so on. The problem that may occur is that this may continue without end, producing an infinite family of central orthogonal idempotents. The condition "R does not contain infinite sets of central orthogonal idempotents" is a type of finiteness condition on the ring. It can be achieved in many ways, such as requiring the ring to be right Noetherian. If a decomposition exists with each "c"i a centrally primitive idempotent, then "R" is a direct sum of the corner rings "c"i"Rc"i, each of which is ring irreducible.
For associative algebras or Jordan algebras over a field, the Peirce decomposition is a decomposition of an algebra as a sum of eigenspaces of commuting idempotent elements.
Relation with involutions.
If "a" is an idempotent of the endomorphism ring End"R"("M"), then the endomorphism is an "R" module involution of "M". That is, "f" is an "R" homomorphism such that "f" 2 is the identity endomorphism of "M".
An idempotent element "a" of "R" and its associated involution "f" gives rise to two involutions of the module "R", depending on viewing "R" as a left or right module. If "r" represents an arbitrary element of "R", "f" can be viewed as a right "R"-homomorphism so that , or "f" can also be viewed as a left "R" module homomorphism , where .
This process can be reversed if 2 is an invertible element of "R": if "b" is an involution, then and are orthogonal idempotents, corresponding to "a" and . Thus for a ring in which 2 is invertible, the idempotent elements correspond to involutions in a one-to-one manner.
Category of "R" modules.
Lifting idempotents also has major consequences for the category of "R" modules. All idempotents lift modulo "I" if and only if every "R" direct summand of "R"/"I" has a projective cover as an "R" module. Idempotents always lift modulo nil ideals and rings for which "R"/"I" is I-adically complete.
Lifting is most important when , the Jacobson radical of "R". Yet another characterization of semiperfect rings is that they are semilocal rings whose idempotents lift modulo J("R").
Lattice of idempotents.
One may define a partial order on the idempotents of a ring as follows: if "a" and "b" are idempotents, we write if and only if . With respect to this order, 0 is the smallest and 1 the largest idempotent. For orthogonal idempotents "a" and "b", is also idempotent, and we have and . The atoms of this partial order are precisely the primitive idempotents.
When the above partial order is restricted to the central idempotents of "R", a lattice structure, or even a Boolean algebra structure, can be given. For two central idempotents "e" and "f" the complement and the join and meet are given by
and
The ordering now becomes simply if and only if , and the join and meet satisfy and . It is shown in that if "R" is von Neumann regular and right self-injective, then the lattice is a complete lattice.


Locally nilpotent derivation

In mathematics, a derivation formula_1 of a commutative ring formula_2 is called a locally nilpotent derivation (LND) if every element of formula_2 is annihilated by some power of formula_1.
One motivation for the study of locally nilpotent derivations comes from the fact that some of the counterexamples to Hilbert's 14th problem are obtained as the kernels of a derivation on a polynomial ring.
Over a field formula_5 of characteristic zero, to give a locally nilpotent derivation on the integral domain formula_2, finitely generated over the field, is equivalent to giving an action of the additive group formula_7 to the affine variety formula_8. Roughly speaking, an affine variety admitting "plenty" of actions of the additive group is considered similar to an affine space.
Definition.
Let formula_2 be a ring. Recall that a derivation of formula_2 is a map formula_11 satisfying the Leibniz rule formula_12 for any formula_13. If formula_2 is an algebra over a field formula_5, we additionally require formula_1 to be formula_5-linear, so formula_18.
A derivation formula_1 is called a "locally nilpotent derivation" (LND) if for every formula_20, there exists a positive integer formula_21 such that formula_22.
If formula_2 is graded, we say that a locally nilpotent derivation formula_1 is "homogeneous" (of degree formula_25) if formula_26 for every formula_27.
The set of locally nilpotent derivations of a ring formula_2 is denoted by formula_29. Note that this set has no obvious structure: it is neither closed under addition (e.g. if formula_30, formula_31 then formula_32 but formula_33, so formula_34) nor under multiplication by elements of formula_2 (e.g. formula_36, but formula_37). However, if formula_38 then formula_39 implies formula_40 and if formula_41, formula_42 then formula_43.
Relation to formula_44-actions.
Let formula_2 be an algebra over a field formula_5 of characteristic zero (e.g. formula_47). Then there is a one-to-one correspondence between the locally nilpotent formula_5-derivations on formula_2 and the actions of the additive group formula_44 of formula_5 on the affine variety formula_52, as follows.
A formula_44-action on formula_52 corresponds to an formula_5-algebra homomorphism formula_56. Any such formula_57 determines a locally nilpotent derivation formula_1 of formula_2 by taking its derivative at zero, namely formula_60 where formula_61 denotes the evaluation at formula_62.
Conversely, any locally nilpotent derivation formula_1 determines a homomorphism formula_56 by formula_65
It is easy to see that the conjugate actions correspond to conjugate derivations, i.e. if formula_66 and formula_41 then formula_68 and formula_69
The kernel algorithm.
The algebra formula_70 consists of the invariants of the corresponding formula_44-action. It is algebraically and factorially closed in formula_2. A special case of Hilbert's 14th problem asks whether formula_70 is finitely generated, or, if formula_74, whether the quotient formula_75 is affine. By Zariski's finiteness theorem, it is true if formula_76. On the other hand, this question is highly nontrivial even for formula_77, formula_78. For formula_79 the answer, in general, is negative. The case formula_80 is open.
However, in practice it often happens that formula_81 is known to be finitely generated: notably, by the Maurer–Weitzenböck theorem, it is the case for "linear" LND's of the polynomial algebra over a field of characteristic zero (by "linear" we mean homogeneous of degree zero with respect to the standard grading).
Assume formula_70 is finitely generated. If formula_83 is a finitely generated algebra over a field of characteristic zero, then formula_81 can be computed using van den Essen's algorithm, as follows. Choose a "local slice", i.e. an element formula_85 and put formula_86. Let formula_87 be the "Dixmier map" given by formula_88. Now for every formula_89, chose a minimal integer formula_90 such that formula_91, put formula_92, and define inductively formula_93 to be the subring of formula_2 generated by formula_95. By induction, one proves that formula_96 are finitely generated and if formula_97 then formula_98, so formula_99 for some formula_100. Finding the generators of each formula_93 and checking whether formula_97 is a standard computation using Gröbner bases.
Slice theorem.
Assume that formula_103 admits a "slice", i.e. formula_104 such that formula_105. The "slice theorem" asserts that formula_2 is a polynomial algebra formula_107 and formula_108.
For any local slice formula_109 we can apply the slice theorem to the localization formula_110, and thus obtain that formula_2 is "locally" a polynomial algebra with a standard derivation. In geometric terms, if a geometric quotient formula_112 is affine (e.g. when formula_76 by the Zariski theorem), then it has a Zariski-open subset formula_114 such that formula_115 is isomorphic over formula_114 to formula_117, where formula_44 acts by translation on the second factor.
However, in general it is not true that formula_119 is locally trivial. For example, let formula_120. Then formula_81 is a coordinate ring of a singular variety, and the fibers of the quotient map over singular points are two-dimensional.
If formula_122 then formula_123 is a curve. To describe the formula_44-action, it is important to understand the geometry formula_125. Assume further that formula_47 and that formula_127 is smooth and contractible (in which case formula_128 is smooth and contractible as well) and choose formula_125 to be minimal (with respect to inclusion). Then Kaliman proved that each irreducible component of formula_125 is a "polynomial curve", i.e. its normalization is isomorphic to formula_131. The curve formula_125 for the action given by Freudenburg's (2,5)-derivation (see below) is a union of two lines in formula_133, so formula_125 may not be irreducible. However, it is conjectured that formula_125 is always contractible.
Examples.
The standard coordinate derivations formula_136 of a polynomial algebra formula_137 are locally nilpotent. The corresponding formula_138-actions are translations: formula_139, formula_140 for formula_141.
Let formula_142, formula_143, and let formula_1 be the Jacobian derivation formula_145. Then formula_146 and formula_147 (see below); that is, formula_1 annihilates no variable. The fixed point set of the corresponding formula_44-action equals formula_150.
Consider formula_151. The locally nilpotent derivation formula_152 of its coordinate ring corresponds to a natural action of formula_138 on formula_154 via right multiplication of upper triangular matrices. This action gives a nontrivial formula_138-bundle over formula_156. However, if formula_47 then this bundle is trivial in the smooth category
LND's of the polynomial algebra.
Let formula_5 be a field of characteristic zero (using Kambayashi's theorem one can reduce most results to the case formula_47
) and let formula_160 be a polynomial algebra.
formula_161 (formula_44-actions on an affine plane).
Every LND of formula_163 can be conjugated to formula_164 for some formula_165. This result is closely related to the fact that every automorphism of an affine plane is tame, and does not hold in higher dimensions.
formula_166 (formula_44-actions on an affine 3-space).
The kernel of every nontrivial LND of formula_168 is isomorphic to a polynomial ring in two variables; that is, a fixed point set of every nontrivial formula_44-action on formula_170 is isomorphic to formula_171.
In other words, for every formula_172 there exist formula_173 such that formula_174 (but, in contrast to the case formula_161, formula_2 is not necessarily a polynomial ring over formula_70). In this case, formula_1 is a Jacobian derivation: formula_145.
Assume that formula_166 and formula_41 is homogeneous relative to some positive grading of formula_2 such that formula_183 are homogeneous. Then formula_184 for some homogeneous formula_185. Moreover, if formula_186 are relatively prime, then formula_187 are relatively prime as well.
A quotient morphism formula_188 of a formula_44-action is surjective. In other words, for every formula_172, the embedding formula_191 induces a surjective morphism formula_192.
This is no longer true for formula_193, e.g. the image of a quotient map formula_194 by a formula_44-action formula_196 (which corresponds to a LND given by formula_197 equals formula_198.
Every fixed-point free action of formula_44 on formula_170 is conjugate to a translation. In other words, every formula_201 such that the image of formula_1 generates the unit ideal (or, equivalently, formula_1 defines a nowhere vanishing vector field), admits a slice. This results answers one of the conjectures from Kraft's list.
Again, this result is not true for formula_193: e.g. consider the formula_205. The points formula_206 and formula_207 are in the same orbit of the corresponding formula_138-action if and only if formula_209; hence the (topological) quotient is not even Hausdorff, let alone homeomorphic to formula_210.
Let formula_103. Then formula_2 is faithfully flat over formula_81. Moreover, the ideal formula_214 is principal in formula_2.
Triangular derivations.
Let formula_216 be any system of variables of formula_2; that is, formula_218. A derivation of formula_2 is called "triangular" with respect to this system of variables, if formula_220 and formula_221 for formula_222. A derivation is called "triangulable" if it is conjugate to a triangular one, or, equivalently, if it is triangular with respect to some system of variables. Every triangular derivation is locally nilpotent. The converse is true for formula_223 by Rentschler's theorem above, but it is not true for formula_224.
The derivation of formula_225 given by formula_226 is not triangulable. Indeed, the fixed-point set of the corresponding formula_44-action is a quadric cone formula_228, while by the result of Popov, a fixed point set of a triangulable formula_44-action is isomorphic to formula_230 for some affine variety formula_231; and thus cannot have an isolated singularity.
The above necessary geometrical condition was later generalized by Freudenburg. To state his result, we need the following definition:
A corank of formula_41 is a maximal number formula_233 such that there exists a system of variables formula_234 such that formula_235. Define formula_236 as formula_21 minus the corank of formula_1.
We have formula_239 and formula_240 if and only if in some coordinates, formula_241 for some formula_242.
Theorem: If formula_41 is triangulable, then any hypersurface contained in the fixed-point set of the corresponding formula_44-action is isomorphic to formula_245.
In particular, LND's of maximal rank formula_21 cannot be triangulable. Such derivations do exist for formula_224: the first example is the "(2,5)-homogeneous derivation" (see above), and it can be easily generalized to any formula_224.
Makar-Limanov invariant.
The intersection of the kernels of all locally nilpotent derivations of the coordinate ring, or, equivalently, the ring of invariants of all formula_44-actions, is called "Makar-Limanov invariant" and is an important algebraic invariant of an affine variety. For example, it is trivial for an affine space; but for the Koras–Russell cubic threefold, which is diffeomorphic to formula_210, it is not.


Arthur Byron Coble

Arthur Byron Coble (November 3, 1878 – December 8, 1966) was an American mathematician. He did research on finite geometries and the group theory related to them, Cremona transformations associated with the Galois theory of equations, and the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface. He was President of the American Mathematical Society from 1933 to 1934.
Biography.
Early life.
Arthur Coble was born on November 3, 1878, in Williamstown, Pennsylvania. His mother Emma was a schoolteacher. When Coble was born, his father Ruben was the manager of a store. Later, he became president of a bank. Coble's parents belonged to Evangelical Lutheran Church. Coble was brought up strictly as an Evangelical Lutheran; however, he rejected this Church when he reached adulthood.
Coble entered Gettysburg College in 1893, and completed his A.B. in 1897. He spent a year as a public school teacher. He entered Johns Hopkins University in 1898 to pursue his graduate studies. He completed his Ph.D. from the university in 1902. His Ph.D. thesis was "The Relation of the Quartic Curve to Conics". His thesis supervisor was English-born mathematician Frank Morley. Later, Coble recalled how Morley made it "a cardinal point to have on hand a sufficient variety of thesis problems to accommodate particular tastes and capacities."
Academic career.
In 1902, Coble became an instructor in mathematics at the University of Missouri. One year later, in 1903, he was appointed to Johns Hopkins University as Morley's research assistant. In 1903, he published his doctoral dissertation as "The quartic curve as related to conics" in the "Transactions of the American Mathematical Society" and took up the research assistant position in Baltimore, Maryland. In 1902, American businessman Andrew Carnegie founded the Carnegie Institution of Washington. The research of Coble and Morley were one of the first pieces of research the Institution supported. The funding of the Institute was generous enough to allow Coble to use the grant to travel abroad. He traveled to Germany where he studied at Greifswald University and the University of Bonn. He wanted to work with Eduard Study, who was well known to mathematicians at Johns Hopkins University because he had taught there in 1893.
Coble returned to the United States for the start of the 1904-05 session. He was appointed an instructor in mathematics at Johns Hopkins University.
Coble married Abby Walker Adams Whitney in 1905. They had four children.
Coble was promoted to associate professor at Johns Hopkins University in 1909. He left Johns Hopkins after he was offered a full professorship at the University of Illinois at Urbana-Champaign (UIUC) in 1918. He remained at Illinois for the rest of his career. He was a visiting professor at the University of Chicago in 1919 and was at Johns Hopkins University in 1927–28. He became head of the Department of Mathematics at the UIUC in 1934 and he held that position until his retirement in 1947. During these years, Coble served on many university and college committees, including eleven years on the University Council and eight years on the Executive Committee of the UIUC College of Liberal Arts and Sciences.
American Mathematical Society.
Coble was active with the American Mathematical Society (AMS) from 1912 to 1940. He was vice-president of the AMS in 1917. From 1920 to 1925, he edited the "Transactions of the American Mathematical Society". He also was involved with editing the "American Journal of Mathematics" over many years between 1918 and 1933. From 1933 to 1934, he was President of the AMS. At that time, the AMS was in some financial difficulties. Coble dealt with the problem effectively.
Later life.
By the time he retired in 1947 his health was already deteriorating due to Parkinson's disease. After his retirement, he accepted a one-year post at Haverford College but after teaching for one semester he resigned due to poor health. In 1956, he was involved in a car crash. Because of that crash, he was unable to walk without assistance. He then moved to Lykens, Pennsylvania, and spent his final ten years of his life there. He died on December 8, 1966, in a hospital in Harrisburg, Pennsylvania.
Research.
Early mathematical research papers written by Coble when he was teaching at Johns Hopkins University, include: "On the relation between the three-parameter groups of a cubic space curve and a quadric surface" (1906); "An application of the form-problems associated with certain Cremona groups to the solution of equations of higher degree" (1908); "An application of Moore's cross-ratio group to the solution of the sextic equation" (1911); "An application of finite geometry to the characteristic theory of the odd and even theta functions" (1913); and "Point sets and allied Cremona groups" (1915).
Coble was interested in finite geometries and the related group theory, and in the Cremona transformations related to the Galois theory of equations. Later in his career, Coble also studied the relations between hyperelliptic theta functions, irrational binary invariants, the Weddle surface and the Kummer surface.
Coble published the monograph "Algebraic geometry and theta functions" in the tenth volume of "American Mathematical Society Colloquium Publications" in 1929, and it was republished by the American Mathematical Society in 1961 and 1982.
Coble published "Configurations defined by theta functions", which reviewed the invariant theory of Cremona transformations as developed by Coble in his earlier papers, in the "Duke Mathematical Journal" in 1939. A linear homogeneous transformation with integral coefficients is associated with a Cremona transformation. These transformations form a group, which Coble studied.
In 1940, Coble published "Trilinear forms" in the "Duke Mathematical Journal". In 1946, he published "Ternary and quaternary elimination", which extends work by mathematicians Francis Sowerby Macaulay and Bartel Leendert van der Waerden, and also extends work done by Frank Morley and Coble some 20 years earlier.


Kravchuk polynomials

Kravchuk polynomials or Krawtchouk polynomials (also written using several other transliterations of the Ukrainian surname "Кравчу́к") are discrete orthogonal polynomials associated with the binomial distribution, introduced by .
The first few polynomials are (for "q"=2):
The Kravchuk polynomials are a special case of the Meixner polynomials of the first kind.
Definition.
For any prime power "q" and positive integer "n", define the Kravchuk polynomial
Properties.
The Kravchuk polynomial has the following alternative expressions:
Symmetry relations.
For integers formula_8, we have that
Orthogonality relations.
For non-negative integers "r", "s",
Generating function.
The generating series of Kravchuk polynomials is given as below. Here formula_11 is a formal variable.


Geoff Smith (mathematician)

Geoffrey Charles Smith, MBE (born 1953) is a British mathematician. He is Senior Lecturer in Mathematics at the University of Bath (where he works in group theory) and current professor in residence at Wells Cathedral School.
He was educated at Trinity School in Croydon, and attended Keble College, Oxford, the University of Warwick, and the University of Manchester, where he gained a Ph.D. in group theory in 1983.
Smith was the leader of the United Kingdom team at the International Mathematical Olympiad between 2002 and 2010, a longer continuous period than any other person. He returned to the position as leader of the British Mathematical Olympiad from 2013.
Smith oversaw a quantitative increase in training: annual events in Bath (moving to The Queen's College, Oxford from 2009), at Oundle School, in Hungary, at Trinity College, Cambridge, and immediately prior to the IMO itself. He also thrice won the IMO Golden Microphone, awarded to the national team leader who makes the most speeches to the IMO Jury. In 2010 he was elected to the IMO Advisory Board for a four-year period. Smith was elected as the chair of the International Mathematical Olympiad for the term of 2014-2018 and was re-elected in 2018.
Smith also prepared UK teams for the Romanian Masters in Mathematics tournament (which they won in 2008), and for participation as guests at the annual Balkan Mathematical Olympiad.
As well as group theory, he is also interested in Euclidean geometry. He often collaborates with Christopher Bradley and David Monk, and has published several papers on Forum Geometricorum, the online geometry journal.
In June 2011 Smith was awarded an MBE for services to education following his contributions toward organising Royal Institution Maths Masterclasses.


Small cancellation theory

In the mathematical subject of group theory, small cancellation theory studies groups given by group presentations satisfying small cancellation conditions, that is where defining relations have "small overlaps" with each other. Small cancellation conditions imply algebraic, geometric and algorithmic properties of the group. Finitely presented groups satisfying sufficiently strong small cancellation conditions are word hyperbolic and have word problem solvable by Dehn's algorithm. Small cancellation methods are also used for constructing Tarski monsters, and for solutions of Burnside's problem.
History.
Some ideas underlying the small cancellation theory go back to the work of Max Dehn in the 1910s. Dehn proved that fundamental groups of closed orientable surfaces of genus at least two have word problem solvable by what is now called Dehn's algorithm. His proof involved drawing the Cayley graph of such a group in the hyperbolic plane and performing curvature estimates via the Gauss–Bonnet theorem for a closed loop in the Cayley graph to conclude that such a loop must contain a large portion (more than a half) of a defining relation.
A 1949 paper of Tartakovskii was an immediate precursor for small cancellation theory: this paper provided a solution of the word problem for a class of groups satisfying a complicated set of combinatorial conditions, where small cancellation type assumptions played a key role. The standard version of small cancellation theory, as it is used today, was developed by Martin Greendlinger in a series of papers in the early 1960s, who primarily dealt with the "metric" small cancellation conditions. In particular, Greendlinger proved that finitely presented groups satisfying the "C"′(1/6) small cancellation condition have word problem solvable by Dehn's algorithm. The theory was further refined and formalized in the subsequent work of Lyndon, Schupp and Lyndon-Schupp, who also treated the case of non-metric small cancellation conditions and developed a version of small cancellation theory for amalgamated free products and HNN-extensions.
Small cancellation theory was further generalized by Alexander Ol'shanskii who developed a "graded" version of the theory where the set of defining relations comes equipped with a filtration and where a defining relator of a particular grade is allowed to have a large overlap with a defining relator of a higher grade. Olshaskii used graded small cancellation theory to construct various "monster" groups, including the Tarski monster and also to give a new proof that free Burnside groups of large odd exponent are infinite (this result was originally proved by Adian and Novikov in 1968 using more combinatorial methods).
Small cancellation theory supplied a basic set of examples and ideas for the theory of word-hyperbolic groups that was put forward by Gromov in a seminal 1987 monograph "Hyperbolic groups".
Main definitions.
The exposition below largely follows Ch. V of the book of Lyndon and Schupp.
Pieces.
Let
be a group presentation where "R" ⊆ "F"("X") is a set of freely reduced and cyclically reduced words in the free group "F"("X") such that "R" is "symmetrized", that is, closed under taking cyclic permutations and inverses.
A nontrivial freely reduced word "u" in "F"("X") is called a "piece" with respect to (∗) if there exist two distinct elements "r"1, "r"2 in "R" that have "u" as maximal common initial segment.
Note that if formula_2 is a group presentation where the set of defining relators "S" is not symmetrized, we can always take the "symmetrized closure" "R" of "S", where "R" consists of all cyclic permutations of elements of "S" and "S"−1. Then "R" is symmetrized and formula_3 is also a presentation of "G".
Metric small cancellation conditions.
Let 0 < "λ" < 1. Presentation (∗) as above is said to satisfy the "C"′("λ") "small cancellation condition" if whenever "u" is a piece with respect to (∗) and "u" is a subword of some "r" ∈ "R", then |"u"| < "λ"|"r"|. Here |"v"| is the length of a word "v".
The condition "C"′("λ") is sometimes called a "metric small cancellation condition".
Non-metric small cancellation conditions.
Let "p" ≥ 3 be an integer. A group presentation (∗) as above is said to satisfy the "C"("p") "small cancellation condition" if whenever "r" ∈ "R" and
where "u""i" are pieces and where the above product is freely reduced as written, then "m" ≥ "p". That is, no defining relator can be written as a reduced product of fewer than "p" pieces.
Let "q" ≥ 3 be an integer. A group presentation (∗) as above is said to satisfy the T("q") "small cancellation condition" if whenever 3 ≤ t < "q" and "r"1...,"r"t in "R" are such that "r"1 ≠ "r"2−1...,
"r"t ≠ "r"1−1 then at least one of the products "r"1"r"2...,"r"t−1"r"t, "r"t"r"1 is freely reduced as written.
Geometrically, condition T("q") essentially means that if "D" is a reduced van Kampen diagram over (∗) then every interior vertex of "D" of degree at least three actually has degree at least "q".
Basic results of small cancellation theory.
Greendlinger's lemma.
The main result regarding the metric small cancellation condition is the following statement (see Theorem 4.4 in Ch. V of ) which is usually called
Greendlinger's lemma:
Let (∗) be a group presentation as above satisfying the "C"′("λ") small cancellation condition where 0 ≤ "λ" ≤ 1/6. Let "w" ∈ "F"("X") be a nontrivial freely reduced word such that "w" = 1 in "G". Then there is a subword "v" of "w" and a defining relator "r" ∈ "R" such that "v" is also a subword of "r" and such that
Note that the assumption "λ" ≤ 1/6 implies that  (1 − 3"λ") ≥ 1/2, so that "w" contains a subword more than a half of some defining relator.
Greendlinger's lemma is obtained as a corollary of the following geometric statement:
Under the assumptions of Greendlinger's lemma, let "D" be a reduced van Kampen diagram over (∗) with a cyclically reduced boundary label such that "D" contains at least two regions. Then there exist two distinct regions "D"1 and "D"2 in "D" such that for "j" = 1,2 the region "D""j" intersects the boundary cycle ∂"D" of "D" in a simple arc whose length is bigger than (1 − 3"λ")|∂"D""j"|.
This result in turn is proved by considering a dual diagram for "D". There one defines a combinatorial notion of curvature (which, by the small cancellation assumptions, is negative at every interior vertex), and one then obtains a combinatorial version of the Gauss–Bonnet theorem. Greendlinger's lemma is proved as a consequence of this analysis and in this way the proof evokes the ideas of the original proof of Dehn for the case of surface groups.
Dehn's algorithm.
For any symmetrized group presentation (∗), the following abstract procedure is called Dehn's algorithm:
Note that we always have
which implies that the process must terminate in at most |"w"| steps. Moreover, all the words "w""j" represent the same element of "G" as does "w" and hence if the process terminates with the empty word, then "w" represents the identity element of "G".
One says that for a symmetrized presentation (∗) "Dehn's algorithm solves the word problem in" "G" if the converse is also true, that is if for any freely reduced word "w" in "F"("X") this word represents the identity element of "G" if and only if Dehn's algorithm, starting from "w", terminates in the empty word.
Greendlinger's lemma implies that for a "C"′(1/6) presentation Dehn's algorithm solves the word problem.
If a "C"′(1/6) presentation (∗) is finite (that is both "X" and "R" are finite), then Dehn's algorithm is an actual non-deterministic algorithm in the sense of recursion theory. However, even if (∗) is an infinite "C"′(1/6) presentation, Dehn's algorithm, understood as an abstract procedure, still correctly decides whether or not a word in the generators "X"±1 represents the identity element of "G".
Asphericity.
Let (∗) be a "C"′(1/6) or, more generally, C(6) presentation where every "r" ∈ "R" is not a proper power in "F"("X") then "G" is aspherical in the following sense. Consider a minimal subset "S" of "R" such that the symmetrized closure of "S" is equal to "R". Thus if "r" and "s" are distinct elements of "S" then "r" is not a cyclic permutation of "s"±1 and formula_10 is another presentation for "G". Let "Y" be the presentation complex for this presentation. Then (see and Theorem 13.3 in ), under the above assumptions on (∗), "Y" is a classifying space for "G", that is "G" = "π"1("Y") and the universal cover of "Y" is contractible. In particular, this implies that "G" is torsion-free and has cohomological dimension two.
More general curvature.
More generally, it is possible to define various sorts of local "curvature" on any van Kampen diagram to be - very roughly - the average excess of (which, by Euler's formula, must total 2) and, by showing, in a particular group, that this is always non-positive (or – even better – negative) internally, show that the curvature must all be on or near the boundary and thereby try to obtain a solution of the word problem. Furthermore, one can restrict attention to diagrams that do not contain any of a set of "regions" such that there is a "smaller" region with the same boundary.
Applications.
Examples of applications of small cancellation theory include:


Affine q-Krawtchouk polynomials

In mathematics, the affine "q"-Krawtchouk polynomials are a family of basic hypergeometric orthogonal polynomials in the basic Askey scheme, introduced by Carlitz and Hodges. give a detailed list of their properties.
Definition.
The polynomials are given in terms of basic hypergeometric functions by
Relation to other polynomials.
affine q-Krawtchouk polynomials → little q-Laguerre polynomials：


J-structure

In mathematics, a J-structure is an algebraic structure over a field related to a Jordan algebra. The concept was introduced by to develop a theory of Jordan algebras using linear algebraic groups and axioms taking the Jordan inversion as basic operation and Hua's identity as a basic relation. There is a classification of simple structures deriving from the classification of semisimple algebraic groups. Over fields of characteristic not equal to 2, the theory of J-structures is essentially the same as that of Jordan algebras.
Definition.
Let "V" be a finite-dimensional vector space over a field "K" and "j" a rational map from "V" to itself, expressible in the form "n"/"N" with "n" a polynomial map from "V" to itself and "N" a polynomial in "K"["V"]. Let "H" be the subset of GL("V") × GL("V") containing the pairs ("g","h") such that "g"∘"j" = "j"∘"h": it is a closed subgroup of the product and the projection onto the first factor, the set of "g" which occur, is the "structure group" of "j", denoted "G"'("j").
A "J-structure" is a triple ("V","j","e") where "V" is a vector space over "K", "j" is a birational map from "V" to itself and "e" is a non-zero element of "V" satisfying the following conditions.
The "norm" associated to a J-structure ("V","j","e") is the numerator "N" of "j", normalised so that "N"("e") = 1. The "degree" of the J-structure is the degree of "N" as a homogeneous polynomial map.
The "quadratic map" of the structure is a map "P" from "V" to End("V") defined in terms of the differential d"j" at an invertible "x". We put
The quadratic map turns out to be a quadratic polynomial map on "V".
The subgroup of the structure group "G" generated by the invertible quadratic maps is the "inner structure group" of the J-structure. It is a closed connected normal subgroup.
J-structures from quadratic forms.
Let "K" have characteristic not equal to 2. Let "Q" be a quadratic form on the vector space "V" over "K" with associated bilinear form "Q"("x","y") = "Q"("x"+"y") − "Q"("x") − "Q"("y") and distinguished element "e" such that "Q"("e".) is not trivial. We define a reflection map "x"* by
and an inversion map "j" by
Then ("V","j","e") is a J-structure.
Example.
Let "Q" be the usual sum of squares quadratic function on "K""r" for fixed integer "r", equipped with the standard basis "e"1...,"e""r". Then ("K""r", "Q", "e""r") is a J-structure of degree 2. It is denoted O2.
Link with Jordan algebras.
In characteristic not equal to 2, which we assume in this section, the theory of J-structures is essentially the same as that of Jordan algebras.
Let "A" be a finite-dimensional commutative non-associative algebra over "K" with identity "e". Let "L"("x") denote multiplication on the left by "x". There is a unique birational map "i" on "A" such that "i"("x")."x" = "e" if "i" is regular on "x": it is homogeneous of degree −1 and an involution with "i"("e") = "e". It may be defined by "i"("x") = "L"("x")−1."e". We call "i" the "inversion" on "A".
A Jordan algebra is defined by the identity
An alternative characterisation is that for all invertible "x" we have
If "A" is a Jordan algebra, then ("A","i","e") is a J-structure. If ("V","j","e") is a J-structure, then there exists a unique Jordan algebra structure on "V" with identity "e" with inversion "j".
Link with quadratic Jordan algebras.
In general characteristic, which we assume in this section, J-structures are related to quadratic Jordan algebras. We take a quadratic Jordan algebra to be a finite dimensional vector space "V" with a quadratic map "Q" from "V" to End("V") and a distinguished element "e". We let "Q" also denote the bilinear map "Q"("x","y") = "Q"("x"+"y") − "Q"("x") − "Q"("y"). The properties of a quadratic Jordan algebra will be
We call "Q"("x")"e" the "square" of "x". If the squaring is dominant (has Zariski dense image) then the algebra is termed "separable".
There is a unique birational involution "i" such that "Q"("x")"i" "x" = "x" if "Q" is regular at "x". As before, "i" is the "inversion", definable by "i"("x") = "Q"("x")−1 "x".
If ("V","j","e") is a J-structure, with quadratic map "Q" then ("V","Q","e") is a quadratic Jordan algebra. In the opposite direction, if ("V","Q","e") is a separable quadratic Jordan algebra with inversion "i", then ("V","i","e") is a J-structure.
H-structure.
McCrimmon proposed a notion of "H-structure" by dropping the density axiom and strengthening the third (a form of Hua's identity) to hold in all isotopes. The resulting structure is categorically equivalent to a quadratic Jordan algebra.
Peirce decomposition.
A J-structure has a Peirce decomposition into subspaces determined by idempotent elements. Let "a" be an idempotent of the J-structure ("V","j","e"), that is, "a"2 = "a". Let "Q" be the quadratic map. Define
This is invertible for non-zero "t","u" in "K" and so φ defines a morphism from the algebraic torus GL1 × GL1 to the inner structure group "G"1. There are subspaces
and these form a direct sum decomposition of "V". This is the Peirce decomposition for the idempotent "a".
Generalisations.
If we drop the condition on the distinguished element "e", we obtain "J-structures without identity". These are related to isotopes of Jordan algebras.