# Finitely generated simple group

From Groupprops

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: finitely generated group and simple group

View other group property conjunctions OR view all group properties

## Definition

A **finitely generated simple group** is a nontrivial group that is both a finitely generated group (i.e., it has a finite generating set) and a [[simple group] (i.e., it has no proper nontrivial normal subgroup).

This includes the groups of prime order (which are the only simple abelian groups), the finite simple non-abelian groups, and the infinite finitely generated simple groups, which must be abelian. There are infinitely many isomorphism classes of groups of each type.