# Even automorphism

## 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.

## Facts

• For $n = 3,4,5$, the outer automorphisms of the alternating group $A_n$ are not even.
• Consider the projective special linear group $PSL(2,q)$ as a subgroup of the projective general linear group $PGL(2,q)$ for an odd prime power $q$. It turns out that the outer automorphisms of $PSL(2,q)$ induced by conjugation by elements of $PGL(2,q)$ are not even.
• For $n \ge 6$, the outer automorphisms of $A_n$ induced by conjugation by odd permutations in $S_n$ are even.