2025/04/29 by V. S. Atabekyan, Atabekyan, V. S., Arman A. Bayramyan +3
Mathematics · #Finite Group Theory Research #Geometric and Algebraic Topology #Advanced Operator Algebra Research
paper · pdf · doi:10.48550/arxiv.2504.20591
In this paper we construct a continuum family of non-isomorphic 3-generator groups in which the identity xn = 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 [xpn, ypn]n = 1. We prove that in each of these products, the probability of satisfying xn = 1 is equal to 1, despite the fact that the identity does not hold throughout any of these groups.