Infinite simple group

From Groupprops
Jump to: navigation, search
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: infinite group and simple group
View other group property conjunctions OR view all group properties

Definition

A group is termed an infinite simple group if it satisfies the following equivalent conditions:

  1. It is an infinite group (i.e., its underlying set has infinitely many elements, or, its order is infinite) as well as a simple group (it has no proper nontrivial normal subgroup).
  2. It is an infinite group as well as a simple non-abelian group. Note that the non-abelianness follows because the only simple abelian groups are cyclic of prime order, and hence finite.

Infinite simple groups can be contrasted with finite simple groups, which could either be cyclic of prime order or finite simple non-abelian.

Examples