Symmetric groups are almost simple

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 (?)).

Statement

For finite sets

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

For infinite sets

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

Definitions used

Almost simple group

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

Facts used

1. Alternating group is normal in symmetric group
2. Alternating groups are simple: The alternating group on $n$ letters, for $n \ge 5$, is a simple group.
3. Finitary alternating group is centralizer-free in symmetric group

Proof

For finite sets

For $n \ge 5$, let $G = S_n$ and $N = A_n$ be the subgroup comprising the even permutations, i.e., the alternating group. Then, we have:

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

Thus, $G$ is an almost simple group.