The Prime Graphs of Some Classes of Finite Groups

TL;DR

This paper characterizes the prime graphs of various classes of finite groups, extending previous work and exploring connections to conjectures, with an algorithm to reconstruct prime graphs from dual graphs.

Contribution

It provides complete characterizations of prime graphs for several classes of finite groups, including square-free, metanilpotent, and cube-free groups, and introduces an algorithm for graph reconstruction.

Findings

Characterizations of prime graphs for specific group classes.

Connections to Maslova's conjecture.

An algorithm for prime graph reconstruction.

Abstract

In this paper we study prime graphs of finite groups. The prime graph of a finite group $G$, also known as the Gruenberg-Kegel graph, is the graph with vertex set {primes dividing $|G|$} and an edge $p$-$q$ if and only if there exists an element of order $pq$ in $G$. In finite group theory, studying the prime graph of a group has been an important topic for the past almost half century. Only recently prime graphs of solvable groups have been characterized in graph theoretical terms only. In this paper, we continue this line of research and give complete characterizations of several classes of groups, including groups of square-free order, metanilpotent groups, groups of cube-free order, and, for any $n\in \mathbb{N}$, solvable groups of $n^\text{th}$-power-free order. We also explore the prime graphs of groups whose composition factors are cyclic or $A_5$ and draw connections to a conjecture of Maslova. We then propose an algorithm that recovers the prime graph from a dual prime graph.