Positive laws on large sets of generators: counterexamples for infinitely generated groups

TL;DR

This paper demonstrates that the finite generation condition is essential in certain group law transfer results, providing a counterexample for infinitely generated groups where previous theorems applied.

Contribution

It constructs a counterexample showing that the finite generation assumption cannot be removed from existing positive law transfer theorems in group theory.

Findings

Counterexample for infinitely generated groups

Finite generation is necessary for law transfer results

Challenges assumptions in residually finite p-groups

Abstract

Shumyatsky and the second author proved that if G is a finitely generated residually finite p-group satisfying a law, then, for almost all primes, the fact that a normal and commutator-closed set of generators satisfies a positive law implies that the whole of G also satisfies a (possibly different) positive law. In this paper, we construct a counterexample showing that the hypothesis of finite generation of the group G cannot be dispensed with.