Symmetric groups on finite sets are complete

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

• Symmetric groups on infinite sets are complete
• Automorphism group of alternating group equals symmetric group: This again holds under the same assumptions: $n \ne 2,6$.

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.