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