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
.