Editing Group (mathematics) (section)

== History ==
{{Main|History of group theory}}
The modern concept of an [[abstract group]] developed out of several fields of mathematics.{{sfn|Wussing|2007}}{{sfn|Kleiner|1986}}{{sfn|Smith|1906}} The original motivation for group theory was the quest for solutions of [[polynomial equation]]s of degree higher than 4. The 19th-century French mathematician [[Évariste Galois]], extending prior work of [[Paolo Ruffini (mathematician)|Paolo Ruffini]] and [[Joseph-Louis Lagrange]], gave a criterion for the [[equation solving|solvability]] of a particular polynomial equation in terms of the [[symmetry group]] of its [[root of a function|roots]] (solutions). The elements of such a [[Galois group]] correspond to certain [[permutation]]s of the roots. At first, Galois's ideas were rejected by his contemporaries, and published only posthumously.{{sfn|Galois|1908}}{{sfn|Kleiner|1986|p=202}} More general permutation groups were investigated in particular by [[Augustin Louis Cauchy]]. [[Arthur Cayley]]'s ''On the theory of groups, as depending on the symbolic equation <math>\theta^n=1</math>'' (1854) gives the first abstract definition of a [[finite group]].{{sfn|Cayley|1889}}

Geometry was a second field in which groups were used systematically, especially symmetry groups as part of [[Felix Klein]]'s 1872 [[Erlangen program]].{{sfn|Wussing|2007|loc=§III.2}} After novel geometries such as [[hyperbolic geometry|hyperbolic]] and [[projective geometry]] had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, [[Sophus Lie]] founded the study of [[Lie group]]s in 1884.{{sfn|Lie|1973}}

The third field contributing to group theory was [[number theory]]. Certain abelian group structures had been used implicitly in [[Carl Friedrich Gauss]]'s number-theoretical work ''[[Disquisitiones Arithmeticae]]'' (1798), and more explicitly by [[Leopold Kronecker]].{{sfn|Kleiner|1986|p=204}} In 1847, [[Ernst Kummer]] made early attempts to prove [[Fermat's Last Theorem]] by developing [[class group|groups describing factorization]] into [[prime number]]s.{{sfn|Wussing|2007|loc=§I.3.4}}

The convergence of these various sources into a uniform theory of groups started with [[Camille Jordan]]'s {{lang|fr|Traité des substitutions et des équations algébriques}} (1870).{{sfn|Jordan|1870}} [[Walther von Dyck]] (1882) introduced the idea of specifying a group by means of generators and relations, and was also the first to give an axiomatic definition of an "abstract group", in the terminology of the time.{{sfn|von Dyck|1882}} As of the 20th century, groups gained wide recognition by the pioneering work of [[Ferdinand Georg Frobenius]] and [[William Burnside]] (who worked on [[representation theory]] of finite groups), [[Richard Brauer]]'s [[modular representation theory]] and [[Issai Schur]]'s papers.{{sfn|Curtis|2003}} The theory of Lie groups, and more generally [[locally compact group]]s was studied by [[Hermann Weyl]], [[Élie Cartan]] and many others.{{sfn|Mackey|1976}} Its [[algebra]]ic counterpart, the theory of [[algebraic group]]s, was first shaped by [[Claude Chevalley]] (from the late 1930s) and later by the work of [[Armand Borel]] and [[Jacques Tits]].{{sfn|Borel|2001}}

The [[University of Chicago]]'s 1960–61 Group Theory Year brought together group theorists such as [[Daniel Gorenstein]], [[John G. Thompson]] and [[Walter Feit]], laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the [[classification of finite simple groups]], with the final step taken by [[Michael Aschbacher|Aschbacher]] and Smith in 2004. This project exceeded previous mathematical endeavours by its sheer size, in both length of [[mathematical proof|proof]] and number of researchers. Research concerning this classification proof is ongoing.{{sfn|Solomon|2018}} Group theory remains a highly active mathematical branch,{{efn|The [[MathSciNet]] database of mathematics publications lists 1,779 research papers on group theory and its generalizations written in 2020 alone. See {{harvnb|MathSciNet|2021}}.}} impacting many other fields, as the [[#Examples and applications|examples below]] illustrate.