Symmetric groups are almost simple

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.

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

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.