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

Kenta Kasai

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.

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

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 . Applied to residue rings, this yields an exact classification: contains the pattern if and only if at least two prime-power factors of exceed 2 . We also compare this phenomenon with the permutation-group setting, where the same pattern is easy to realize.

Paper Structure

This paper contains 8 sections, 10 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Affine permutations over commutative rings
  3. The local obstruction
  4. Product constructions
  5. Consequences for residue rings
  6. Comparison with permutation groups
  7. Conclusion
  8. Coding-theoretic motivation

Key Result

Lemma 2.1

Two affine permutations $(a,b)$ and $(c,d)$ in $\mathrm{AGL}_1(R)$ commute if and only if $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (24)

  • Lemma 2.1: Commutation criterion
  • proof
  • Definition 2.2: Cross-commuting nonabelian square
  • Theorem 3.1: Local obstruction
  • proof
  • Corollary 3.2: Family version
  • proof
  • Theorem 4.1: Product construction
  • proof
  • Proposition 4.2: Product family construction
  • ...and 14 more