Symmetric groups are almost simple

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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.