On non-normal subgroup perfect codes

TL;DR

This paper constructs an infinite family of finite groups with non-normal subgroup perfect codes, answering a question about the existence of such codes with specific algebraic properties.

Contribution

It provides the first known infinite family of finite groups with non-normal subgroup perfect codes exhibiting particular element square properties.

Findings

Existence of non-normal subgroup perfect codes in certain finite groups.

Construction of an infinite family of such groups.

Answer to an open question by Wang, Xia, and Zhou.

Abstract

Let $X = (V,E)$ be a graph. A subset $C \subseteq V(X)$ is a \emph{perfect code} of $X$ if $C$ is a coclique of $X$ with the property that any vertex in $V(X)\setminus C$ is adjacent to exactly one vertex in $C$. Given a finite group $G$ with identity element $e$ and $H\leq G$, $H$ is a \emph{subgroup perfect code} of $G$ if there exists an inverse-closed subset $S \subseteq G\setminus \{e\}$ such that $H$ is a perfect code of the Cayley graph $\operatorname{Cay}(G,S)$ of $G$ with connection set $S$. In this short note, we give an infinite family of finite groups $G$ admitting a non-normal subgroup perfect code $H$ such that there exists $ g\in G$ with $g^2\in H$ but $(gh)^2 \neq e$, for all $h \in H$; thus, answering a question raised by Wang, Xia, and Zhou in [Perfect sets in Cayley graphs. {\it arXiv preprint} arXiv:2006.05100, 2020].