# Alternating group implies every element is automorphic to its inverse

From Groupprops

## Contents

## Statement

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

## Related facts

### Related facts about alternating groups

- Alternating group implies any two elements generating the same cyclic subgroup are automorphic
- Classification of ambivalent alternating groups: is ambivalent for precisely the values .
- Classification of alternating groups having a class-inverting automorphism: has a class-inverting automorphism for precisely the values .

### Related facts about every element being automorphic to its inverse

- General linear group implies every element is automorphic to its inverse
- Projective general linear group implies every element is automorphic to its inverse
- Special linear group implies every element is automorphic to its inverse
- Projective special linear group implies every element is automorphic to its inverse
- Every element is automorphic to its inverse is characteristic subgroup-closed
- Normal subgroup of ambivalent group implies every element is automorphic to its inverse

## Facts used

## Proof

Consider the group , in which is a normal subgroup of index two. Since is a rational group, the elements and are conjugate. Thus, there exists such that .

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