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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2604.02063 |
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Table of 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.