Characterizing Finite Groups via Subgroup Perfect Codes

TL;DR

This paper investigates the relationship between subgroup perfect codes in finite groups and the number of prime divisors of the group order, providing classifications for specific cases.

Contribution

It establishes bounds on the number of conjugacy classes of subgroup perfect codes relative to prime divisors and classifies groups attaining these bounds.

Findings

Proved that |elta(G)|  |(G)| with only three exceptions.

Classified finite groups where |elta(G)| equals |(G)| or |(G)| + 1.

Characterized all insoluble groups with at most 6 subgroup perfect codes.

Abstract

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.