标题： 非同构3-生成元群的一连续统具有概率律 $x^n=1$ 显示英文标题

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

摘要： 本文中，我们构造了一个连续族的非同构的3生成群，在这些群中，恒等式$x^n = 1$以概率1成立，而在每个群中普遍满足该恒等式的条件却不成立。 这解决了关于群恒等式概率满足和普遍满足之间关系的一个近期问题。 我们的构造使用了$n$周期乘积的循环群，其阶数为$n$，以及两个生成元且满足形如$[x^{pn}, y^{pn}]^n = 1$的恒等式的相对自由群。 我们证明，在这些乘积中的每一个中，满足$x^n = 1$的概率等于1，尽管该恒等式并不在这类群的任何一个中普遍成立。 显示英文摘要

摘要： 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.