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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.20591 |
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| _version_ | 1866914071051239424 |
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| author | Atabekyan, V. S. Bayramyan, A. A. Mikaelian, V. H. |
| author_facet | Atabekyan, V. S. Bayramyan, A. A. Mikaelian, V. H. |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_20591 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A continuum of non-isomorphic 3-generator groups with probabilistic law $x^n=1$ Atabekyan, V. S. Bayramyan, A. A. Mikaelian, V. H. Group Theory 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. |
| title | A continuum of non-isomorphic 3-generator groups with probabilistic law $x^n=1$ |
| topic | Group Theory |
| url | https://arxiv.org/abs/2504.20591 |