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.