Divisibility graph for symmetric and alternating groups

Abstract

Let $X$ be a non-empty set of positive integers and $X^{\ast }=X\setminus \{1\}$. The divisibility graph $D(X)$ has $X^{\ast }$ as the vertex set and there is an edge connecting $a$ and $b$ with $a,b\in X^{\ast }$ whenever $a$ divides $b$ or $b$ divides $a$. Let $X=csG$ be the set of conjugacy class sizes of a group $G$. In this case, we denote $D(csG)$ 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 $A_n$.

Cited by in corpus (1)

• On divisibility graph for simple Zassenhaus groups