Cross-Commuting Nonabelian Squares in Affine Groups over Finite Commutative Principal Ideal Rings

TL;DR

This paper investigates a specific commutation pattern in affine groups over finite rings, revealing conditions under which the pattern can or cannot occur, with implications for group theory and permutation groups.

Contribution

It provides a classification of when the nonabelian commuting pattern occurs in affine groups over finite principal ideal rings, especially relating to prime factorization of the ring.

Findings

The pattern is impossible over local principal ideal rings with noncommuting affine permutations.

The pattern exists in direct product rings with noncommuting pairs separated into different factors.

The pattern occurs in $ ext{AGL}_1( ext{Z}/n ext{Z})$ if and only if at least two prime-power factors of n are greater than 2.

Abstract

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.