Varieties of finite supersolvable groups with the M. Hall property

摘要

The varieties in the title are shown to be precisely the product varieties G p *Ab(d) for some prime p and some positive integer d dividing p−1. Here G p denotes the variety of all finite p-groups and Ab(d) the variety of all finite Abelian groups of exponent dividing d. It turns out that these are exactly those varieties H of supersolvable groups for which all finitely generated free pro-H groups are freely indexed in the sense of Lubotzky and van den Dries. Several alternative characterizations of these varieties are presented. Some applications to formal language theory and finite monoid theory are also given. Among these is the determination of all supersolvable solutions H to the equations PH = J*H and J*H = J H which is, to the present date, the most complete solution to a problem raised by Pin. Another consequence of our results is that for each such variety H the monoid variety PH = J*H = J H has decidable membership.