Symmetric groups on finite sets are complete

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Statement

For n \ne 2,6, the Symmetric group (?) \operatorname{Sym}(n) on a set of size n (i.e., the Symmetric group on finite set (?)), is a Complete group (?): it is centerless and every automorphism of it is inner.

For n = 2, the group is not centerless, but every automorphism is inner.

For n = 6, the group is centerless, but not every automorphism is inner. In fact, the symmetric group of degree six is of index two in its automorphism group.

Related facts

Facts used

  1. Symmetric groups are centerless (for the centerlessness part)
  2. Conjugacy class of transpositions is preserved by automorphisms
  3. Transposition-preserving automorphism of symmetric group is inner

Proof

  1. Centerless: The fact that the symmetric group is centerless for n \ne 2 follows from fact (1).
  2. Every automorphism is inner: Fact (2) yields that every automorphism preserves the conjugacy class of transpositions when n \ne 6 follows from fact (2). Fact (3) then yields that, in fact, every automorphism is inner.