Group properties characterised by configurations

J. M. Rosenblatt and G. A. Willis introduced the notion of configurations for finitely generated groups G. They characterised amenability of G in terms of the configuration equations. In this paper we investigate which group properties can be characterised by configurations. It is proved that if G₁ and G₂ are two finitely generated groups having the same configuration sets and G₁ satisfies a semigroup law, then G₂ satisfies the same semigroup law. Furthermore, if G₁ is abelian then G₁ and G₂ are isomorphic.