# Subgroup perfect codes in Cayley graphs

## TL;DR

This paper characterizes groups where every proper subgroup is a perfect code in some Cayley graph, showing this occurs if and only if the group has no elements of order 4, and explores subgroup perfect codes in abelian and quaternion groups.

## Contribution

It provides a complete characterization of code-perfect groups and reduces the problem of subgroup perfect codes in abelian groups to abelian 2-groups, also classifying subgroup perfect codes in quaternion groups.

## Key findings

- A group is code-perfect iff it has no elements of order 4.
- Subgroup perfect codes in abelian groups depend on Sylow 2-subgroups.
- All subgroup perfect codes in generalized quaternion groups are determined.

## Abstract

Let $\Gamma$ be a graph with vertex set $V(\Gamma)$. A subset $C$ of $V(\Gamma)$ is called a perfect code in $\Gamma$ if $C$ is an independent set of $\Gamma$ and every vertex in $V(\Gamma)\setminus C$ is adjacent to exactly one vertex in $C$. A subset $C$ of a group $G$ is called a perfect code of $G$ if there exists a Cayley graph of $G$ which admits $C$ as a perfect code. A group $G$ is said to be code-perfect if every proper subgroup of $G$ is a perfect code of $G$. In this paper we prove that a group is code-perfect if and only if it has no elements of order $4$. We also prove that a proper subgroup $H$ of an abelian group $G$ is a perfect code of $G$ if and only if the Sylow $2$-subgroup of $H$ is a perfect code of the Sylow $2$-subgroup of $G$. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian $2$-groups. Finally, we determine all subgroup perfect codes in any generalized quaternion group.