A Characterization of the Prime Graphs of Solvable Groups

TL;DR

This paper characterizes prime graphs of solvable groups, showing they are precisely those whose complements are 3-colorable and triangle-free, and explores properties of minimal prime graphs and their implications.

Contribution

It provides a complete characterization of prime graphs of solvable groups and introduces the concept of minimal prime graphs with new structural results.

Findings

Prime graphs of solvable groups are characterized by complement being 3-colorable and triangle-free.

Existence of an infinite class of solvable groups with minimal prime graphs.

Proof of the 3k-conjecture for solvable groups with minimal prime graphs.

Abstract

Let \pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G is the graph with vertex set \pi(G) with edges {p,q} if and only if there exists an element of order pq in G. In this paper, we prove that a graph is isomorphic to the prime graph of a solvable group if and only if its complement is 3-colorable and triangle free. We then introduce the idea of a minimal prime graph. We prove that there exists an infinite class of solvable groups whose prime graphs are minimal. We prove the 3k-conjecture on prime divisors in element orders for solvable groups with minimal prime graphs, and we show that solvable groups whose prime graphs are minimal have Fitting length 3 or 4.