The lower central series of groups of a special class

Abstract

The structure of the lower central series of free groups of finite rank is well known ([l], [7]). To describe the factor groups of the lower central series of an arbitrary group is, however, very difficult. Recently this has been done for one-relator groups ([5], [lo]). Such factor groups are important in Burnside’s problem ([3], [4]). It seems desirable to obtain results for other groups. In this paper we shall study the factor groups arising from groups, G, of a special class through the use of basic commutators. We shall assume that G is a free product of finitely many groups, G(i), and that every G(i) is the direct product of finitely many groups of prime order. The factor groups are then either of order one, or they are direct products of finitely many groups of prime order, just as the groups G(i). We shall discuss the computation of the number of groups of a given prime order in such a direct product. For this purpose we shall introduce special sets of basic commutators all of which have the same interesting structure.