A Characterization of the Prime Graphs of Solvable Groups

2013/05/10 by Gruber, Alexander, Keller, Thomas, Lewis, Mark +2 · 5 citations
#FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.1305.2368

Abstract

Let π(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 π(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.

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• Degree 2 vertices in minimal prime graph complements
• Realizing Gruenberg-Kegel graphs of T-solvable groups with structurally simplified extensions of T
• Criteria for Classifying Prime Graphs of PSL(2, q)-Solvable Groups
• On the Codegree graphs of finite groups
• Prime simplicial complexes of finite groups

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