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.

Stronger properties

 * Weaker than::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.

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.