Cross-Commuting Nonabelian Squares in Affine Groups over Finite Commutative Principal Ideal Rings
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866912998867599360 |
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| author | Kasai, Kenta |
| author_facet | Kasai, Kenta |
| contents | We study a commutation pattern in which two affine families commute completely across the two families while each family retains internal noncommutativity. For one-dimensional affine groups over finite commutative rings, we prove a local-product dichotomy. Over a finite commutative local principal ideal ring, the common centralizer of two noncommuting affine permutations is always abelian, so the pattern is impossible. Over a direct product of two commutative rings whose affine groups each contain a noncommuting pair, the same pattern is constructed by separating the two noncommuting families into different factors. More generally, over a finite commutative principal ideal ring, the pattern exists if and only if at least two local factors are not isomorphic to $\mathbb{F}_2$. Applied to residue rings, this yields an exact classification: $\mathrm{AGL}_1(\mathbb{Z} / n \mathbb{Z})$ contains the pattern if and only if at least two prime-power factors of $n$ exceed 2 . We also compare this phenomenon with the permutation-group setting, where the same pattern is easy to realize. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_02063 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Cross-Commuting Nonabelian Squares in Affine Groups over Finite Commutative Principal Ideal Rings Kasai, Kenta Group Theory We study a commutation pattern in which two affine families commute completely across the two families while each family retains internal noncommutativity. For one-dimensional affine groups over finite commutative rings, we prove a local-product dichotomy. Over a finite commutative local principal ideal ring, the common centralizer of two noncommuting affine permutations is always abelian, so the pattern is impossible. Over a direct product of two commutative rings whose affine groups each contain a noncommuting pair, the same pattern is constructed by separating the two noncommuting families into different factors. More generally, over a finite commutative principal ideal ring, the pattern exists if and only if at least two local factors are not isomorphic to $\mathbb{F}_2$. Applied to residue rings, this yields an exact classification: $\mathrm{AGL}_1(\mathbb{Z} / n \mathbb{Z})$ contains the pattern if and only if at least two prime-power factors of $n$ exceed 2 . We also compare this phenomenon with the permutation-group setting, where the same pattern is easy to realize. |
| title | Cross-Commuting Nonabelian Squares in Affine Groups over Finite Commutative Principal Ideal Rings |
| topic | Group Theory |
| url | https://arxiv.org/abs/2604.02063 |