# Even automorphism

From Groupprops

## Definition

An automorphism of a finite group is termed an **even** automorphism if it is an even permutation.

For any finite group, the even automorphisms form a subgroup of the group of all automorphisms. This subgroup is either the whole group, or is a subgroup of index two.

Note that the definition is vacuous over infinite groups since an infinite group has no non-identity finitary automorphism.

## Relation with other properties

### Stronger properties

- Inner automorphism: In a finite group, every inner automorphism is even. Thus, we can talk of whether an element in the outer automorphism group is even or not.
`For full proof, refer: Inner implies even in finite group`

## Facts

- For , the outer automorphisms of the alternating group are
*not*even. - Consider the projective special linear group as a subgroup of the projective general linear group for an odd prime power . It turns out that the outer automorphisms of induced by conjugation by elements of are
*not*even. - For , the outer automorphisms of induced by conjugation by odd permutations in are even.