Extreme classes of finite soluble groups

Abstract

Knowledge of the structure of a specified collection 9 of subgroups of a group G can often yield detailed information about G itself. For example we have the elementary fact that if all the minimal cyclic subgroups of a group G are n-groups then G is a n-group. Another example is provided by an unpublished theorem of M. B. Powell who has shown that if we take for 9’ the collection of 2-generator subgroups of a finite soluble group G then G has p-length at most n if and only if every member of Y has p-length at most n. Clearly one should aim to make the set Y as small as possible consistent with still retrieving useful information about the structure of G. In this note we investigate a situation of this type and for the purpose introduce a class (3 of extreme groups. (5 is a subclass of the class of finite soluble groups and loosely speaking comprises those groups which contain as few complemented chief factors, as possible, in a given chief series; more precisely, a group G is extreme if it has a chief series with exactly Z(G) complemented chief factors, where Z(G) denotes the nilpotent (or Fitting) length of G. @ is a subclass of the class 8, of 2-generator groups, and contains, for example, cyclic p-groups but no other cyclic groups. To avoid introducing at this stage the unfamiliar notation which gives the main theorem its full generality we content ourselves here with a statement of some of its more interesting consequences: