The Prime Index Graph of a Group

Abstract

Let $G$ be a group. The prime index graph of $G$, denoted by $\Pi (G)$, is the graph whose vertex set is the set of all subgroups of $G$ and two distinct comparable vertices $H$ and $K$ are adjacent if and only if the index of $H$ in $K$ or the index of $K$ in $H$ is prime. In this paper, it is shown that for every group $G$, $\Pi (G)$ is bipartite and the girth of $\Pi (G)$ is contained in the set $\{4,\infty \}$. Also we prove that if $G$ is a finite solvable group, then $\Pi (G)$ is connected.

Cited by in corpus (1)

• The topology of local commensurability graphs