Automorphism group of alternating group equals symmetric group

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Statement

For n \ne 2,3,6, the automorphism group of the alternating group \operatorname{Alt}(n), i.e., the alternating group of degree n, is isomorphic to the symmetric group of degree n, with the action given by conjugation in the symmetric group.

In other words, the automorphism tower of the alternating group of degree n stabilizes after one step, at the symmetric group of degree n.

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