# Symmetric groups on finite sets are complete

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## Contents

## Statement

For , the Symmetric group (?) on a set of size (i.e., the Symmetric group on finite set (?)), is a Complete group (?): it is centerless and every automorphism of it is inner.

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

For , 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

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

## Proof

- Centerless: The fact that the symmetric group is centerless for follows from fact (1).
- 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.