# Alternating group implies every element is automorphic to its inverse

## 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 $g^{-1}$.

## 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 $A_n$.