Editing Group theory (section)

==History==
{{Main|History of group theory}}
Group theory has three main historical sources: [[number theory]], the theory of [[algebraic equation]]s, and [[geometry]]. The number-theoretic strand was begun by [[Leonhard Euler]], and developed by [[Carl Friedrich Gauss|Gauss's]] work on [[modular arithmetic]] and additive and multiplicative groups related to [[quadratic field]]s. Early results about permutation groups were obtained by [[Joseph Louis Lagrange|Lagrange]], [[Paolo Ruffini (mathematician)|Ruffini]], and [[Niels Henrik Abel|Abel]] in their quest for general solutions of polynomial equations of high degree. [[Évariste Galois]] coined the term "group" and established a connection, now known as [[Galois theory]], between the nascent theory of groups and [[field theory (mathematics)|field theory]]. In geometry, groups first became important in [[projective geometry]] and, later, [[non-Euclidean geometry]]. [[Felix Klein]]'s [[Erlangen program]] proclaimed group theory to be the organizing principle of geometry.

[[Évariste Galois|Galois]], in the 1830s, was the first to employ groups to determine the solvability of [[polynomial equation]]s. [[Arthur Cayley]] and [[Augustin Louis Cauchy]] pushed these investigations further by creating the theory of permutation groups. The second historical source for groups stems from [[geometry|geometrical]] situations. In an attempt to come to grips with possible geometries (such as [[euclidean geometry|euclidean]], [[hyperbolic geometry|hyperbolic]] or [[projective geometry]]) using group theory, [[Felix Klein]] initiated the [[Erlangen programme]]. [[Sophus Lie]], in 1884, started using groups (now called [[Lie group]]s) attached to [[analysis (mathematics)|analytic]] problems. Thirdly, groups were, at first implicitly and later explicitly, used in [[algebraic number theory]].

The different scope of these early sources resulted in different notions of groups. The theory of groups was unified starting around 1880. Since then, the impact of group theory has been ever growing, giving rise to the birth of [[abstract algebra]] in the early 20th century, [[representation theory]], and many more influential spin-off domains. The [[classification of finite simple groups]] is a vast body of work from the mid 20th century, classifying all the [[finite set|finite]] [[simple group]]s.