Probabilistic Identities in n-Torsion Groups

In this paper, a 3-generated infinite group is constructed in which the probability of satisfying a certain fixed identity tends to 1. At the same time, this identity is not satisfied throughout the entire group. The question of the existence of such a group was recently posed by several authors. More precisely, the work considers an \(n\) -periodic product of a free periodic group of rank 2 and an infinite cyclic group. It is proved that in this product, the probability of satisfying the identity \(x^{n}=1\) tends to 1, but it does not hold on the entire product.