Alternating group implies any two elements generating the same cyclic subgroup are automorphic

Statement

Let $n$ be a natural number. Then, the alternating group of degree $n$ has the property that any two elements generating the same cyclic subgroup are automorphic.

Facts used

1. Symmetric groups are rational
2. Normal subgroup of rational group implies any two elements generating the same cyclic subgroup are automorphic

Proof

The proof follows from facts (1) and (2), and the fact that the alternating group of degree $n$ is a normal subgroup inside the symmetric group of degree $n$.