On w-maximal groups

Let w=w(x1,…,xn) be a word, i.e. an element of the free group F=〈x1,…,xn〉 on n generators x1,…,xn. The verbal subgroup w(G) of a group G is the subgroup generated by the set of all w-values in G. We say that a (finite) group G is w-maximal if |G:w(G)|>|H:w(H)| for all proper subgroups H of G and that G is hereditarily w-maximal if every subgroup of G is w-maximal. In this text we study w-maximal and hereditarily w-maximal (finite) groups.