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