On the absolute centre and the autocommutator subgroup of a group

Let G be a group. The autocentre, or absolute centre, of G is the subgroup of G of all the elements fixed by every automorphism of G,while the autocommutator subgroup of G is the subgroup of G generated by all the elements g⁻¹g^(α), with g ∈ G and α ∈ Aut(G). In the present article we introduce some homological tools to study the properties of these subgroups and, by means of these, we extend several results for finite and infinite groups. In particular, we show that the torsion subgroup of the autocentre of G lies almost always inside the Frattini subgroup of G and give polynomial bounds for the order of the autocommutator subgroup of G, improving those previously given by Hegarty.