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

Research paper by **H. Kim**

Indexed on: **30 May '17**Published on: **30 May '17**Published in: **arXiv - Mathematics - Group Theory**

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#### 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 $G$. 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 $H_2 \cap \ker \varphi$. We fully categorize local conjugacy
in $\text{GL}_2(\mathbb{Z}/p^2\mathbb{Z})$ through such casework.