The Prime Index Graph of a Group

TL;DR

This paper introduces the prime index graph of a group, explores its bipartite structure and girth properties, and proves connectivity for finite solvable groups, revealing new insights into subgroup relationships.

Contribution

It establishes fundamental properties of the prime index graph, including bipartiteness, girth constraints, and connectivity in finite solvable groups, advancing understanding of subgroup index structures.

Findings

$ ext{Pi}(G)$ is bipartite for all groups

Girth of $ ext{Pi}(G)$ is either 4 or infinite

Connectivity holds for finite solvable groups

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.