On the autocommutator subgroup and absolute centre of a group

We show that if the quotient of a group by its absolute centre is locally finite of exponent n, then the exponent of its autocommutator subgroup is n-bounded, that is, bounded by a function depending only on n. If the group itself is locally finite, then its exponent is n-bounded as well. Under some extra assumptions, the exponent of its automorphism group is n-bounded. We determine the absolute centre and autocommutator subgroup for a large class of (infinite) abelian groups.