# Infinite simple group

From Groupprops

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:

- 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).
- 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

- The finitary alternating group on any infinite set is simple. See finitary alternating groups are simple.
- The projective special linear group of degree two or higher on an infinite field is simple. See projective special linear groups are simple. There are also other infinite simple groups of Lie type.
- Any group with two conjugacy classes, other than cyclic group:Z2, is an infinite simple group.
- The Tarski monsters are examples of infinite simple groups.