The structure of free prosupersolvable groups

Abstract

Oltikar [9], and Oltikar and Ribes [lo], have shown that the p-Sylow subgroups of a finitely generated prosupersolvable group are finitely generated. The argument is topological and gives no estimate on the number of generators of the subgroups. In Section 1 we prove a theorem on finite groups which gives such a bound for supersolvable groups. The nature of the bound is such that it holds for prosupersolvable groups via a limit argument. Oltikar 191, also showed, using a cohomologicai argument of Gruenberg (p. 164 of [5]), that the p-Sylow subgroups of a free prosupersolvable group are free pro-p. The argument provides no construction of a set of free generators. In Section 2 we construct, for a given set of primes n, and index set, I, a profinite group, F, and subset, D, indexed by I. The construction is by an iterated semidirect product of free pro-p subgroups, F,, on explicitly given generators and with explicitly given actions of F, on the generators of F, for p > q. We show that D is a set of generators for F and that F is prosupersolvable, which, since the bounds of Section 1 are attained, shows the latter are best possible. In Section 3 we establish a mapping property of F relative to D which shows it is “the” free prosupersolvable n-group on D, thus giving a fairly explicit structure theory, including a Hall system, for the latter object, defined by this mapping property. In particular this provides a noncohomological proof of Oltikar’s second result. In Section 4 we use the Hall system to get similar descriptions (iterated semidirect products of known groups with known actions) of certain canonical subgroups of F. In Section 5 we derive some facts about its automorphism group. For basic facts about profinite groups, in particular the notion of (supernatural) order and p-Sylow subgroups, see [ 11, 131. For the notion of free pro-C product and free pro-C group see [ 1, 3, 5, 7, 141. For extending basic facts about supersolvable groups to prosupersolvable see [9, lo]; for the 256 0021-8693/83 $3.00