Editing Group theory (section)

{{Short description|Branch of mathematics that studies the properties of groups}}
{{Hatnote|This article covers advanced notions. For basic topics, see [[Group (mathematics)]].}}
{{For|group theory in social sciences|Social group}}
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[[Image:Rubik's cube.svg|thumb|The popular [[Rubik's Cube]] puzzle, invented in 1974 by [[Ernő Rubik]], has been used as an illustration of [[permutation group]]s. See [[Rubik's Cube group]].]]
In [[abstract algebra]], '''group theory''' studies the [[algebraic structure]]s known as [[group (mathematics)|groups]].  
The concept of a group is central to abstract algebra: other well-known algebraic structures, such as [[ring (mathematics)|rings]], [[field (mathematics)|fields]], and [[vector space]]s, can all be seen as groups endowed with additional [[operation (mathematics)|operation]]s and [[axiom]]s. Groups recur throughout mathematics, and the methods of group theory have influenced many parts of algebra. [[Linear algebraic group]]s and [[Lie group]]s are two branches of group theory that have experienced advances and have become subject areas in their own right.

Various physical systems, such as [[crystal]]s and the [[hydrogen atom]], and [[Standard Model|three of the four]] known fundamental forces in the universe, may be modelled by [[symmetry group]]s. Thus group theory and the closely related [[representation theory]] have many important applications in [[physics]], [[chemistry]], and [[materials science]]. Group theory is also central to [[public key cryptography]].

The early [[history of group theory]] dates from the 19th century. One of the most important mathematical achievements of the 20th century<ref>{{citation|last=Elwes|first=Richard|url=http://plus.maths.org/issue41/features/elwes/index.html|title=An enormous theorem: the classification of finite simple groups|journal=[[Plus Magazine]]|issue=41|date=December 2006|access-date=2011-12-20|archive-date=2009-02-02|archive-url=https://web.archive.org/web/20090202092008/http://plus.maths.org/issue41/features/elwes/index.html|url-status=dead}}</ref> was the collaborative effort, taking up more than 10,000 journal pages and mostly published between 1960 and 2004, that culminated in a complete [[classification of finite simple groups]].