Alternating group implies every element is automorphic to its inverse

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Statement

Let n be any natural number. Let A_n denote the Alternating group (?) of degree n, i.e., the group of even permutations on a set of size n. Then, A_n is a Group in which every element is automorphic to its inverse (?). In other words, for every element g \in A_n, there is an automorphism of A_n that conjugates g to g^{-1}.

Related facts

Related facts about alternating groups

Related facts about every element being automorphic to its inverse

Facts used

  1. Symmetric groups are rational, or cycle type determines conjugacy class

Proof

Consider the group S_n, in which A_n is a normal subgroup of index two. Since S_n is a rational group, the elements g and g^{-1} are conjugate. Thus, there exists x \in S_n such that xgx^{-1} = g^{-1}.

Since A_n is a normal subgroup of S_n, conjugation by x restricts to an automorphism of A_n.