Groupprops, The Group Properties Wiki (pre-alpha)

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From Groupprops

Contents 1 Statement 2 Related facts 3 Facts used 4 Proof

Statement

For , the symmetric group 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: .

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 follows from fact (1).
2. Every automorphism is inner: Fact (2) yields that every automorphism preserves the conjugacy class of transpositions when follows from fact (2). Fact (3) then yields that, in fact, every automorphism is inner.