Characterizing Finite Groups via Subgroup Perfect Codes

Abstract

A perfect code in a graph $\hat{I}“=(V,E)$ is a subset $C$ of $V$ such that no two vertices in $C$ are adjacent and every vertex in $V\setminus C$ is adjacent to exactly one vertex in $C$. A subgroup $H$ of a group $G$ is called a subgroup perfect code of $G$ if it is a perfect code in some Cayley graph of $G$. In this paper, we study the set $\hat{I}”(G)$ of conjugacy classes of nontrivial subgroup perfect codes of $G$, with a focus on its relation to $|\ddot{I}€(G)|$, the number of prime divisors of $|G|$. We prove that $|\hat{I}”(G)|\ge |\ddot{I}€(G)|$ with only three exceptional families, which leads to the natural question: when is this bound attained or nearly attained? We completely classify finite groups $G$ satisfying $|\hat{I}”(G)|=|\ddot{I}€(G)|$ and $|\hat{I}”(G)|=|\ddot{I}€(G)|+1$, and we further characterize all insolvable groups with $|\hat{I}”(G)|\le 6$. Our approach is based on the classification of primitive groups of odd degree, as well as the classification of primitive groups of square-free degree.