# Local Conjugacy in $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$

## TL;DR

This paper fully characterizes local conjugacy among subgroups of GL_2(Z/p^2Z) for odd primes p, building on prior classifications over Z/pZ and analyzing subgroup intersections and images.

## Contribution

It provides a complete classification of local conjugacy in GL_2(Z/p^2Z), extending previous work on GL_2(Z/pZ) and detailing subgroup interactions.

## Key findings

- Characterized conditions for local conjugacy in GL_2(Z/p^2Z)
- Identified subgroup intersection and image structures for conjugacy classification
- Developed casework approach for full categorization

## Abstract

Subgroups $H_1$ and $H_2$ of a group $G$ are said to be locally conjugate if there is a bijection $f: H_1 \rightarrow H_2$ such that $h$ and $f(h)$ are conjugate in $G$ for every $h \in H_1$. This paper studies local conjugacy among subgroups of $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$, where $p$ is an odd prime, building on Sutherland's categorizations of subgroups of $\text{GL}_2(\mathbb{Z}/p\mathbb{Z})$ and local conjugacy among them. There are two conditions that locally conjugate subgroups $H_1$ and $H_2$ of $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ must satisfy: letting $\varphi: \text{GL}_2(\mathbb{Z}/p^2\mathbb{Z}) \rightarrow \text{GL}_2(\mathbb{Z}/p\mathbb{Z})$ be the natural homomorphism, $H_1 \cap \ker \varphi$ and $H_2 \cap \ker \varphi$ must be locally conjugate in $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ and $\varphi(H_1)$ and $\varphi(H_2)$ must be locally conjugate in $\text{GL}_2(\mathbb{Z}/p\mathbb{Z})$. To identify $H_1$ and $H_2$ up to conjugation, we choose $\varphi(H_1)$ and $\varphi(H_2)$ to be similar to each other, then understand the possibilities for $H_1 \cap \ker \varphi$ and $H_2 \cap \ker \varphi$. This study fully categorizes local conjugacy in $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ through such casework.