# Symmetric groups are almost simple

From Groupprops

This article gives the statement, and possibly proof, of a particular group or type of group (namely, Symmetric group (?)) satisfying a particular group property (namely, Almost simple group (?)).

## Contents

## Statement

### For finite sets

Let be a finite set of size at least five. Then the symmetric group on is an almost simple group.

### For infinite sets

Let be an infinite set. Then, both the finitary symmetric group on and the whole symmetric group on are almost simple groups.

## Definitions used

### Almost simple group

A group is termed an **almost simple group** if has a normal subgroup such that is a simple non-Abelian group and is centralizer-free in .

## Facts used

- Alternating group is normal in symmetric group
- Alternating groups are simple: The alternating group on letters, for , is a simple group.
- Finitary alternating group is centralizer-free in symmetric group

## Proof

### For finite sets

For , let and be the subgroup comprising the even permutations, i.e., the alternating group. Then, we have:

- is normal in (by fact (1)).
- is simple and non-Abelian (by fact (2)).
- is centralizer-free (By fact (3)).

Thus, is an almost simple group.