Abstract
It does not seem easy to improve on Theorem 1. In fact there is no complete analog of Schreier's theorem even for the free groups of product varieties. For example, suppose that G is the free metabelian group on a and b. Let N be the normal closure of a 2 and b 2. Then it is not difficult to show that G/N is the infinite dihedral group. However, as we see from Theorem 1' below, N is finitely generated. To each metabelian variety one can associate an integer, termed its
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