A continuum of non-isomorphic 3-generator groups with probabilistic law $x^n=1$

Abstract

In this paper we construct a continuum family of non-isomorphic 3-generator groups in which the identity $x^n = 1$ holds with probability 1, while failing to hold universally in each group. This resolves a recent question about the relationship between probabilistic and universal satisfaction of group identities. Our construction uses $n$-periodic products of cyclic groups of order $n$ and two-generator relatively free groups satisfying identities of the form $[x^{pn}, y^{pn}]^n = 1$. We prove that in each of these products, the probability of satisfying $x^n = 1$ is equal to 1, despite the fact that the identity does not hold throughout any of these groups.