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

From Groupprops

## Statement

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

## Facts used

- Symmetric groups are rational
- 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 is a normal subgroup inside the symmetric group of degree .