A variation of the Ito theorem

Let X be a normal subset of a group G such that X⊆AB for some abelian subgroups A and B. Our main result is that X generates a metabelian subgroup (Theorem 1.1). In the case where X=G this is well-known Ito's theorem. From this we deduce that if G is finite and A and B are π-subgroups, then X generates a π-subgroup (Theorem 1.2). We also show that if G is finite and e is a prime-power such that A and B have exponents dividing e, then the exponent of 〈X〉 is e-bounded (Theorem 1.3).