Editing Simple group (section)

{{Short description|Group without normal subgroups other than the trivial group and itself}}
{{Group theory sidebar |Basics}}

In [[mathematics]], a '''simple group''' is a nontrivial [[Group (mathematics)|group]] whose only [[normal subgroup]]s are the [[trivial group]] and the group itself.  A group that is not simple can be broken into two smaller groups, namely a nontrivial normal subgroup and the corresponding [[quotient group]]. This process can be repeated, and for [[finite group]]s one eventually arrives at uniquely determined simple groups, by the [[Jordan–Hölder theorem]].

The complete [[classification of finite simple groups]], completed in 2004, is a major milestone in the history of mathematics.