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

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 $φ: \text{GL}_2(\mathbb{Z}/p^2\mathbb{Z}) \rightarrow \text{GL}_2(\mathbb{Z}/p\mathbb{Z})$ be the natural homomorphism, $H_1 \cap \ker φ$ and $H_2 \cap \ker φ$ must be locally conjugate in $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ and $φ(H_1)$ and $φ(H_2)$ 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 $φ(H_1)$ and $φ(H_2)$ to be similar to each other, then understand the possibilities for $H_1 \cap \ker φ$ and $H_2 \cap \ker φ$. This study fully categorizes local conjugacy in $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ through such casework.