Editing Group (mathematics) (section)

=== Classification of finite simple groups ===
{{Main|Classification of finite simple groups}}

[[Computer algebra system]]s have been used to [[List of small groups|list all groups of order up to 2000]].{{efn|[[Up to]] isomorphism, there are about 49 billion groups of order up to 2000. See {{harvnb|Besche|Eick|O'Brien|2001}}.}}
But [[classification theorems|classifying]] all finite groups is a problem considered too hard to be solved.

The classification of all finite ''simple'' groups was a major achievement in contemporary group theory. There are [[List of finite simple groups|several infinite families]] of such groups, as well as 26 "[[sporadic groups]]" that do not belong to any of the families. The largest [[sporadic group]] is called the [[monster group]]. The [[monstrous moonshine]] conjectures, proved by [[Richard Borcherds]], relate the monster group to certain [[modular function]]s.{{sfn|Ronan|2007}}

The gap between the classification of simple groups and the classification of all groups lies in the [[extension problem]].<ref>{{harvnb|Aschbacher|2004|p=737}}.</ref>