Characterizing Finite Groups via Subgroup Perfect Codes

A perfect code in a graph $\Gamma = (V, E)$ 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 $\Delta(G)$ of conjugacy classes of nontrivial subgroup perfect codes of $G$, with a focus on its relation to $|\pi(G)|$, the number of prime divisors of $|G|$. We prove that $|\Delta(G)| \ge |\pi(G)|$ 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 $|\Delta(G)| = |\pi(G)|$ and $|\Delta(G)| = |\pi(G)| + 1$, and we further characterize all insolvable groups with $|\Delta(G)| \le 6$. 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.