Products of groups which contain abelian subgroups of finite index

It is unknown whether every group G = AB which is the product of of two abelian-by-finite subgroups A and B must always have a soluble or even metabelian subgroup of finite index. Here we deal with the special case of this problem when A and B contain abelian subgroups of "small" index, notably of index at most 2. Some recent results on the solubility of such groups are discussed which depend on special calculations involving involutions.