A characterization of the prime graphs of solvable groups

[摘要] Let pi(G) denote the set of prime divisors of the order of a finite group G. The prime graph of G, denoted Gamma(G), is the graph with vertex set pi(G) with edges {p, q} is an element of E(Gamma(G)) 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. (C) 2014 Elsevier Inc. All rights reserved.

[发布日期] 2015-11-15

[效力级别]  Proceedings Paper

[关键词] Prime graphs of finite groups;Sets of element orders;Solvable groups;Frobenius groups;3-colorable graphs