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On the diameter of intersection graphs of finite groups

Melissa Lee, Kamilla Rekvényi

Abstract

The intersection graph $Δ_G$ of a finite group $G$ is a simple graph with vertices the non-trivial proper subgroups of $G$, and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Csákány and Pollák in 1969. In this paper we answer two long-standing open questions posed by Csákány and Pollák concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.

On the diameter of intersection graphs of finite groups

Abstract

The intersection graph of a finite group is a simple graph with vertices the non-trivial proper subgroups of , and an edge between two vertices if their corresponding subgroups intersect non-trivially. These graphs were introduced by Csákány and Pollák in 1969. In this paper we answer two long-standing open questions posed by Csákány and Pollák concerning the diameter of intersection graphs. We prove some necessary conditions for a non-simple group to have an intersection graph of diameter 4. We also construct the first examples of non-simple groups and alternating groups whose intersection graphs have diameter 4.
Paper Structure (4 sections, 7 theorems, 3 equations)

This paper contains 4 sections, 7 theorems, 3 equations.

Table of Contents

  1. Non-simple groups with connected intersection graphs
  2. Proof of Theorem \ref{['thm:nonsimple']}
  3. Intersection graphs of alternating groups with prime degree
  4. Proof of Theorem \ref{['thm:An']}

Key Result

Theorem 1

Suppose $G$ is a non-simple group with connected intersection graph $\Delta_G$. Then $\mathrm{diam}(\Delta_G) \leq 4$ with equality only if $G$ is almost simple with socle $G_0$, $G=\langle G_0, g\rangle$, where $g$ is a diagonal automorphism of odd prime order $p$, and one of the following holds. Moreover, if $G=\mathrm{PGU}_5(4)$, then $\mathrm{diam}(\Delta_G) = 4$.

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5
  • proof
  • Remark 1.1
  • proof
  • ...and 6 more