Divisibility graph for symmetric and alternating groups

TL;DR

This paper investigates the structure of divisibility graphs constructed from conjugacy class sizes of symmetric and alternating groups, specifically determining the number of connected components in these graphs.

Contribution

It provides a novel analysis of the divisibility graph structure for symmetric and alternating groups, identifying the number of connected components.

Findings

Determined the number of connected components of D(G) for S_n and A_n.

Established divisibility graph properties for conjugacy class sizes.

Enhanced understanding of the divisibility relations in group conjugacy classes.

Abstract

Let $X$ be a non-empty set of positive integers and $X^*=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^*$ as the vertex set and there is an edge connecting $a$ and $b$ with $a, b\in X^*$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=cs~{G}$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(cs~{G})$ by $D(G)$. In this paper we will find the number of connected components of $D(G)$ where $G$ is the symmetric group $S_n$ or is the alternating group $A_n$.