Abstract

The group laws which are not satisfied in product-varieties ApA for any prime p will be called, after F. Point, the Milnor laws. Let G be a 2-generator relatively free group defined by a nontrivial law w ≡ 1. We show that this law is the Milnor law if and only if G’/G’’ is finitely generated. Moreover, the properties G’ = G’’ and G’ is finitely or infinitely generated, allow us to split the Milnor laws for three disjoint classes. We describe their properties.

Introduction

We denote by F = _x, y_ the free group of rank 2 and by F∞ the free group on X = ﹛x1, x2, … ﹜. A word w = w(x1, x2, … , xn) is called a law for a group G if w(g1, g2,… , gn) = e for all g1, g2, …, gn in G. The law w can be written as w ≡ 1. We denote [x, y] = x−1y−1xy = x−1xy, [x,0y] = x, and [x,n y] = [[x,n−1 y], y].

A variety of groups is the class of all groups satisfying every law in a given set of laws (see [20] Chapter 1). Each variety is defined by a verbal subgroup V ⊆ F∞ and consists of all groups G satisfying V (G) = ﹛e﹜.

By we denote the variety of all abelian groups of a prime exponent p, by A – the variety of all abelian groups. Nc denotes the variety of nilpotent groups of class ≤ c and Be – the variety of all groups of exponent dividing e.

In the following section we describe different types of laws, such as positive laws, pseudo-abelian laws, R-laws, Milnor laws. The corresponding varieties have similar names. The aim of the paper is to show that the commutator subgroup G’ of a 2- generator free group G in the variety defined by a law w ≡ 1 is responsible for the crucial properties of the law.