A Note on Subgroup Perfect Codes in Cayley Graphs

TL;DR

The paper addresses when a subgroup $H$ of a finite group $G$ can be a perfect code in a Cayley graph Cay$(G,S)$ by deriving a necessary and sufficient condition expressed through $\Omega_1$-relations on the normalizer of the $2$-part $H_2$. The authors prove an equivalence linking local $2$-subgroup data and the global perfect-code property, and apply it to extraspecial $2$-groups and to groups with extraspecial Sylow $2$-subgroups, obtaining complete classifications in these cases. They show that, for extraspecial $2$-groups, $H$ is a perfect code precisely when $H$ is non-abelian, or abelian non-normal, or a maximal abelian subgroup when $G\cong G_{m,1}$. For groups with extraspecial Sylow $2$-subgroups, the paper provides explicit criteria based on the structure of $H_2$ and its normalizer, yielding a detailed characterization of subgroup perfect codes in this broader class.

Abstract

In this paper, we give a necessary and sufficient condition for a subgroup to be a perfect code for finite groups. As an application, we determine all subgroup perfect codes of extraspecial 2-groups and finite groups whose Sylow 2-subgroup is extraspecial.