On group actions on groups and associated series

Abstract

1. Introduction. In this paper we consider Q-groups; that is, Q is a group and we consider groups N endowed with a Q-action, meaning a homomorphism of Q into the group of automorphisms of N. In (4) a lower central Q-series was defined for such a Q-group, N, generalizing the lower central series of a group, and results were obtained relating to the localization of such a series. Since the ideas in that paper were inspired by the homotopical localization theory of nilpotent spaces (see (6)), the main body of results in (4) was concerned with the case in which N is nilpotent, and perhaps also the group Q and the action of Q on N (in the sense that the lower central Q-series terminates after a finite number of steps with the trivial group {1}). We now adopt a broader view-point and only restrict ourselves to the nilpotent case when our results appear to require us to do so; thus the spirit of this paper is much more that of general group theory as presented in [(8), especially ch. VI]. Thus, while there is some overlap of results, the methods used are not the same and many results (for example, Theorem 3·1) are far more general than any obtained in (4). Moreover, the methods also appear to us to be more appropriate in that essential appeal was made in (4) to a sophisticated theorem of Norman Blackburn on nilpotent groups, whereas here we merely use homological methods, the construction of the semidirect product and, in section 4, some very classical facts of the commutator calculus.