Varieties of Metabelian p-Groups of Class b, p+1

Abstract

A large number of results are available on the lattice of subvarieties of the variety of metabelian groups. When considering metabelian p-groups (for odd p), the immediate division is between groups of nilpotency class less than p, and those of class at least p. The first case was dealt with in some detail in [1], and this paper extends the results to the next interesting cases, classes p and p + 1. The main results are stated in Theorems 2 and 4, which give the basis laws for certain varieties, and 3 and 5, which assert the existence of specific generating groups for these varieties, and hence their non-trivial existence. The notation, and the essential parts of the logic, are as in [1]; for the purposes of this paper, the following modification of Lemma 1.1 of [1], and also the ring-of-integer operations used in its proof, are together dubbed the ‘Stirling manipulation’: