A completeness criterion for the common divisor graph on p-regular class sizes

2024/12/12 by Sotomayor, Víctor · 1 citation
#20E45 #FOS: Mathematics #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2412.09083

Abstract

Let G be a finite p-separable group, for some fixed prime p. Let Γ_p(G) be the common divisor graph built on the set of sizes of p-regular conjugacy classes of G: this is the simple undirected graph whose vertices are the class sizes of those non-central elements of G such that p does not divide their order, and two distinct vertices are adjacent if and only if they are not coprime. In this note we prove that if Γ_p(G) is a k-regular graph with k≥ 1, then it is a complete graph with k+1 vertices. We also pose a conjecture regarding the order of products of p-regular elements with coprime conjugacy class sizes, whose validity would enable to drop the p-separability hypothesis.

Cited by

• Groups with conjugacy classes of coprime sizes

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• Groups whose common divisor graph on p-regular classes has diameter three
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• Groups with conjugacy classes of coprime sizes