A Continuum of Nonisomorphic 3-Generator Groups with Probabilistic Lawxⁿ=1

In this paper, we construct a continuum family of nonisomorphic 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.