Editing Finite group (section)

{{short description|Mathematical group based upon a finite number of elements}}
{{Group theory sidebar |Basics}}

In [[abstract algebra]], a '''finite group''' is a [[group (mathematics)|group]] whose [[underlying set]] is [[finite set|finite]].  Finite groups often arise when considering symmetry of [[Symmetry in mathematics|mathematical]] or [[Symmetry (physics)|physical]] objects, when those objects admit just a finite number of structure-preserving transformations.  Important examples of finite groups include [[cyclic group]]s and [[permutation group]]s.

The study of finite groups has been an integral part of [[group theory]] since it arose in the 19th century.  One major area of study has been classification: the [[classification of finite simple groups]] (those with no nontrivial [[normal subgroup]]) was completed in 2004.