group theory wikipedia group theory contents main class group edit permutation group edit matrix group edit transformation group edit abstract group edit group additional structure edit branch group theory edit finite group theory edit representation group edit lie theory edit combinatorial geometric group theory edit connection group symmetry edit applications group theory edit galois theory edit algebraic topology edit algebraic geometry edit algebraic number theory edit harmonic analysis edit combinatorics edit music edit physics edit chemistry material science edit statistical mechanic edit cryptography edit history edit edit note edit reference edit external link edit navigation menu personal tool namespaces variant view search navigation contribute tool print export project language wikipedia free encyclopedia redirect group theory jump navigation jump search branch mathematic study property group article cover advanced notion basic topic group mathematics group theory social science social group popular puzzle rubik cube invent ernő rubik illustration permutation group rubik cube group algebraic structure group theory group theory basic notion subgroup normal subgroup quotient group direct product group homomorphism kernel image direct sum wreath product simple finite infinite continuous multiplicative additive cyclic abelian dihedral nilpotent solvable glossary group theory list group theory topic finite group classification finite simple group cyclic alternate lie type sporadic cauchy theorem lagrange theorem sylow theorem hall theorem p group elementary abelian group frobenius group schur multiplier symmetric group s n klein group v dihedral group d n quaternion group q dicyclic group dic n discrete group lattices integers z free group modular group z z arithmetic group lattice hyperbolic group topological lie group solenoid circle general linear gl n special linear sl n orthogonal o n euclidean e n special orthogonal n unitary u n special unitary su n symplectic sp n g f e e e lorentz poincaré conformal diffeomorphism loop infinite dimensional lie group algebraic group linear algebraic group reductive group abelian variety elliptic curve v t e mathematic abstract algebra group theory study algebraic structure know group concept group central abstract algebra know algebraic structure ring field vector space see group endow additional operation axiom group recur mathematic method group theory influence part algebra linear algebraic group lie group branch group theory experience advance subject area right physical system crystal hydrogen atom model symmetry group group theory closely related representation theory important application physics chemistry material science group theory central public key cryptography early history group theory date 19th century important mathematical achievement 20th century collaborative effort take journal page publish culminate complete classification finite simple group main class group permutation group matrix group transformation group abstract group group additional structure branch group theory finite group theory representation group lie theory combinatorial geometric group theory connection group symmetry applications group theory galois theory algebraic topology algebraic geometry algebraic number theory harmonic analysis combinatorics music physics chemistry material science statistical mechanic cryptography history note references external link main class group edit main article group mathematic range group consider gradually expand finite permutation group special example matrix group abstract group specify presentation generator relation permutation group edit class group undergo systematic study permutation group give set x collection g bijection x know permutation close composition inverse g group act x x consist n element g consist permutation g symmetric group s n general permutation group g subgroup symmetric group x early construction cayley exhibit group permutation group act x g mean left regular representation case structure permutation group study property action correspond set example way prove n alternate group n simple admit proper normal subgroup fact play key role impossibility solve general algebraic equation degree n radical matrix group edit important class group give matrix group linear group g set consist invertible matrix give order n field k close product inverse group act n vector space k n linear transformation action make matrix group conceptually similar permutation group geometry action usefully exploit establish property group g transformation group edit permutation group matrix group special case transformation group group act certain space x preserve inherent structure case permutation group x set matrix group x vector space concept transformation group closely related concept symmetry group transformation group frequently consist transformation preserve certain structure theory transformation group form bridge connect group theory differential geometry long line research originate lie klein consider group action manifold homeomorphism diffeomorphism group discrete continuous abstract group edit group consider stage development group theory concrete have realize number permutation matrix late nineteenth century idea abstract group set operation satisfy certain system axiom begin hold typical way specify abstract group presentation generator relation g s r significant source abstract group give construction factor group quotient group g h group g normal subgroup h class group algebraic number field early example factor group interest number theory group g permutation group set x factor group g h long act x idea abstract group permit worry discrepancy change perspective concrete abstract group make natural consider property group independent particular realization modern language invariant isomorphism class group give property finite group periodic group simple group solvable group explore property individual group seek establish result apply class group new paradigm paramount importance development mathematic foreshadow creation abstract algebra work hilbert emil artin emmy noether mathematician school citation need group additional structure edit important elaboration concept group occur g endow additional structure notably topological space differentiable manifold algebraic variety group operation m multiplication inversion m g g g g h g h g g g g m g compatible structure continuous smooth regular sense algebraic geometry map g topological group lie group algebraic group presence extra structure relate type group mathematical discipline mean tool available study topological group form natural domain abstract harmonic analysis lie group frequently realize transformation group mainstay differential geometry unitary representation theory certain classification question solve general approach resolve special subclass group compact connected lie group completely classified fruitful relation infinite abstract group topological group group γ realize lattice topological group g geometry analysis pertain g yield important result γ comparatively recent trend theory finite group exploit connection compact topological group profinite group example single p analytic group g family quotient finite p order property g translate property finite quotient branch group theory edit finite group theory edit main article finite group twentieth century mathematician investigate aspect theory finite group great depth especially local theory finite group theory solvable nilpotent group citation need consequence complete classification finite simple group achieve mean simple group finite group build know second half twentieth century mathematician chevalley steinberg increase understanding finite analog classical group related group family group family general linear group finite field finite group occur consider symmetry mathematical physical object object admit finite number structure preserve transformation theory lie group view deal continuous symmetry strongly influence associate weyl group finite group generate reflection act finite dimensional euclidean space property finite group play role subject theoretical physics chemistry representation group edit main article representation theory say group g act set x mean element g define bijective map set x way compatible group structure x structure useful restrict notion representation g vector space v group homomorphism ρ g gl v gl v gl v consist invertible linear transformation v word group element g assign automorphism ρ g ρ g ρ h ρ gh h g definition understand direction rise new domain mathematic hand yield new information group g group operation g abstractly give ρ correspond multiplication matrix explicit hand give understand group act complicated object simplify study object question example g finite know v decompose irreducible part part turn easily manageable v schur lemma give group g representation theory ask representation g exist setting employ method obtain result different case representation theory finite group representation lie group main subdomain theory totality representation govern group character example fourier polynomial interpret character group complex number absolute value act l periodic function lie theory edit main article lie theory lie group group differentiable manifold property group operation compatible smooth structure lie group name sophus lie lay foundation theory continuous transformation group term groupe de lie appear french thesis lie student arthur tresse page lie group represent well develop theory continuous symmetry mathematical object structure make indispensable tool part contemporary mathematic modern theoretical physics provide natural framework analyse continuous symmetry differential equation differential galois theory way permutation group galois theory analyse discrete symmetry algebraic equation extension galois theory case continuous symmetry group lie principal motivation combinatorial geometric group theory edit main article geometric group theory group describe different way finite group describe write group table consist possible multiplication g h compact way define group generator relation call presentation group give set f generator g free group generate f surject group g kernel map call subgroup relation generate subset d presentation usually denote f d example group presentation b b b describe group isomorphic z z z z string consist generator symbol inverse call word combinatorial group theory study group perspective generator relation particularly useful finiteness assumption satisfied example finitely generate group finitely present group addition relation finite area make use connection graphs fundamental group example subgroup free group free natural question arise give group presentation word problem ask word effectively group element relate problem turing machine general algorithm solve task generally hard algorithmically insoluble problem group isomorphism problem ask group give different presentation actually isomorphic example group presentation x y x y x y x e x xyxyx isomorphic additive group z integer immediately apparent cayley graph x y free group rank geometric group theory attack problem geometric viewpoint view group geometric object find suitable geometric object group act idea precise mean cayley graph vertex correspond group element edge correspond right multiplication group give element construct word metric give length minimal path element theorem milnor svarc say give group g act reasonable manner metric space x example compact manifold g quasi isometric look similar distance space x connection group symmetry edit main article symmetry group give structured object x sort symmetry mapping object preserve structure occur case example x set additional structure symmetry bijective map set give rise permutation group object x set point plane metric structure metric space symmetry bijection set preserve distance pair point isometry corresponding group call isometry group x instead angle preserve speak conformal map conformal map rise kleinian group example symmetry restrict geometrical object include algebraic object instance equation x solution case group exchange root galois group belong equation polynomial equation variable galois group certain permutation group root axiom group formalize essential aspect symmetry symmetry form group close symmetry object apply symmetry result symmetry identity keep object fix symmetry object existence inverse guarantee undo symmetry associativity come fact symmetry function space composition function associative frucht theorem say group symmetry group graph abstract group actually symmetry explicit object saying preserve structure object precise work category map preserve structure morphism symmetry group automorphism group object question application group theory edit applications group theory abound structure abstract algebra special case group ring example view abelian group correspond addition second operation correspond multiplication group theoretic argument underlie large part theory entity galois theory edit main article galois theory galois theory use group describe symmetry root polynomial precisely automorphism algebra generate root fundamental theorem galois theory provide link algebraic field extension group theory give effective criterion solvability polynomial equation term solvability correspond galois group example s symmetric group element solvable imply general quintic equation solve radical way equation low degree theory historical root group theory fruitfully apply yield new result area class field theory algebraic topology edit main article algebraic topology algebraic topology domain prominently associate group object theory interested group describe certain invariant topological space call invariant define way change space subject deformation example fundamental group count path space essentially different poincaré conjecture prove grigori perelman prominent application idea influence unidirectional example algebraic topology make use eilenberg maclane space space prescribed homotopy group similarly algebraic k theory rely way classify space group finally torsion subgroup infinite group show legacy topology group theory torus abelian group structure induce map c c z τ z τ parameter live upper half plane algebraic geometry edit main article algebraic geometry algebraic geometry likewise use group theory way abelian variety introduce presence group operation yield additional information make variety particularly accessible serve test new conjecture dimensional case elliptic curve study particular detail theoretically practically intriguing direction toric variety algebraic variety act torus toroidal embedding recently lead advance algebraic geometry particular resolution singularity algebraic number theory edit main article algebraic number theory algebraic number theory make use group important application example euler product formula n n s p prime p s capture fact integer decompose unique way prime failure statement general ring give rise class group regular prime feature kummer treatment fermat theorem harmonic analysis edit main article harmonic analysis analysis lie group certain group call harmonic analysis haar measure integral invariant translation lie group pattern recognition image processing technique combinatoric edit combinatoric notion permutation group concept group action simplify counting set object particular burnside lemma circle fifth endow cyclic group structure music edit presence periodicity circle fifth yield application elementary group theory musical set theory transformational theory model musical transformation element mathematical group physics edit physics group important describe symmetry law physics obey accord noether theorem continuous symmetry physical system correspond conservation law system physicist interested group representation especially lie group representation point way possible physical theory example use group physics include standard model gauge theory lorentz group poincaré group chemistry material science edit chemistry material science point group classify regular polyhedron symmetry molecule space group classify crystal structure assign group determine physical property chemical polarity chirality spectroscopic property particularly useful raman spectroscopy infrare spectroscopy circular dichroism spectroscopy magnetic circular dichroism spectroscopy uv vis spectroscopy fluorescence spectroscopy construct molecular orbital molecular symmetry responsible physical spectroscopic property compound provide relevant information chemical reaction occur order assign point group give molecule necessary find set symmetry operation present symmetry operation action rotation axis reflection mirror plane word operation move molecule indistinguishable original configuration group theory rotation axis mirror plane call symmetry element element point line plane respect symmetry operation carry symmetry operation molecule determine specific point group molecule water molecule symmetry axis chemistry important symmetry operation identity operation e rotation operation proper rotation c n reflection operation σ inversion rotation reflection operation improper rotation s n identity operation e consist leave molecule equivalent number rotation axi symmetry molecule symmetry group chiral molecule consist identity operation identity operation characteristic molecule symmetry rotation axis c n consist rotate molecule specific axis specific angle rotation angle n n integer rotation axis example water molecule rotate axis pass oxygen atom hydrogen atom configuration start case n apply twice produce identity operation molecule rotation axis cn axi have large value n high order rotation axis principal axi example borane bh3 high order rotation axis c principal axi rotation axis c reflection operation σ molecule mirror plane obvious reflection operation exchange leave right point move perpendicularly plane position exactly far plane start plane perpendicular principal axi rotation call σ h horizontal plane contain principal axi rotation label vertical σ v dihedral σ d inversion complex operation point move center molecule position opposite original position far central point start molecule glance inversion center example methane tetrahedral molecule lack inversion symmetry hold methane model hydrogen atom vertical plane right hydrogen atom horizontal plane left inversion result hydrogen atom horizontal plane right hydrogen atom vertical plane left inversion symmetry operation methane orientation molecule follow inversion operation differ original orientation operation improper rotation rotation reflection operation s n require rotation n follow reflection plane perpendicular axi rotation statistical mechanic edit group theory resolve incompleteness statistical interpretation mechanic develop willard gibbs relate summing infinite number probability yield meaningful solution cryptography edit large group prime order construct elliptic curve cryptography serve public key cryptography cryptographical method kind benefit flexibility geometric object group structure complicated structure group discrete logarithm hard calculate early encryption protocol caesar cipher interpret easy group operation cryptographic scheme use group way particular diffie hellman key exchange use finite cyclic group term group base cryptography refer cryptographic protocol use infinite nonabelian group braid group cyclic group z underlie caesar cipher history edit main article history group theory group theory main historical source number theory theory algebraic equation geometry number theoretic strand begin leonhard euler develop gauss work modular arithmetic additive multiplicative group relate quadratic field early result permutation group obtain lagrange ruffini abel quest general solution polynomial equation high degree évariste galois coin term group establish connection know galois theory nascent theory group field theory geometry group important projective geometry later non euclidean geometry felix klein erlangen program proclaim group theory organize principle geometry galois 1830 employ group determine solvability polynomial equation arthur cayley augustin louis cauchy push investigation create theory permutation group second historical source group stem geometrical situation attempt come grip possible geometry euclidean hyperbolic projective geometry group theory felix klein initiate erlangen programme sophus lie start group call lie group attach analytic problem thirdly group implicitly later explicitly algebraic number theory different scope early source result different notion group theory group unify start impact group theory grow give rise birth abstract algebra early 20th century representation theory influential spin domain classification finite simple group vast body work mid 20th century classify finite simple group edit list group theory topic examples group note edit elwes richard december enormous theorem classification finite simple group plus magazine process impose extra structure formalize notion group object suitable category lie group group object category differentiable manifold affine algebraic group group object category affine algebraic variety group cohomology equivariant k theory particular representation faithful arthur tresse sur les invariant différentiel des groupe continus de transformations pdf acta mathematica doi bf02418270 schupp lyndon write z x y z xy g z y z y z z la harpe example hodge conjecture certain case birch swinnerton dyer conjecture millennium problem abramovich dan karu kalle matsuki kenji wlodarczyk jaroslaw torification factorization birational map journal american mathematical society arxiv doi s0894 mr lenz reiner group theoretical method image processing lecture notes computer science berlin new york springer verlag doi isbn norbert wiener cybernetics control communication animal machine isbn ch reference edit borel armand linear algebraic group graduate texts mathematics 2nd ed berlin new york springer verlag doi isbn mr carter nathan visual group theory classroom resource materials series mathematical association america isbn mr cannon john computer group theory survey communications acm doi mr frucht herstellung von graphen mit vorgegebener abstrakter gruppe compositio mathematica issn 437x archive original golubitsky martin stewart ian nonlinear dynamic network groupoid formalism bull amer math soc doi s0273 mr show advantage generalise group groupoid judson thomas abstract algebra theory application introductory undergraduate text spirit text gallian herstein cover group ring integral domain field galois theory free downloadable pdf open source gfdl license kleiner israel evolution group theory brief survey mathematics magazine doi issn 570x jstor mr la harpe pierre de topic geometric group theory university chicago press isbn livio equation solve mathematical genius discover language symmetry simon schuster isbn convey practical value group theory explain point symmetry physics science mumford david abelian variety oxford university press isbn oclc ronan symmetry monster oxford university press isbn lay reader describe quest find basic building block finite group rotman joseph introduction theory group new york springer verlag isbn standard contemporary reference schupp paul lyndon roger combinatorial group theory berlin new york springer verlag isbn scott group theory new york dover isbn inexpensive fairly readable somewhat dated emphasis style notation shatz stephen profinite group arithmetic geometry princeton university press isbn mr weibel charles introduction homological algebra cambridge studies advanced mathematics cambridge university press isbn mr oclc 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