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Elusive groups from non-split extensions

Jiyong Chen, Melissa Lee, Dorde Mitrovic, E. A. O'Brien, Binzhou Xia

Abstract

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts that no elusive group is $2$-closed. Existing constructions of elusive groups mostly arise from split extensions. In this paper, we initiate the construction of elusive groups via non-split extensions. As a demonstration, we construct elusive groups of new degrees, namely $p^{3k-4}(p+1)/2$ for each Mersenne prime $p\geq7$ and integer $k\geq2$. We also construct the first examples of elusive groups with odd degree, namely $3^{k+1}\cdot5^2$, and twice odd degree, namely $2\cdot3^{k + 1}\cdot5^2$ for each $k\geq1$. We conclude by proposing further problems to advance this new direction of research.

Elusive groups from non-split extensions

Abstract

A finite transitive permutation group is elusive if it contains no derangements of prime order. These groups are closely related to a longstanding open problem in algebraic graph theory known as the Polycirculant Conjecture, which asserts that no elusive group is -closed. Existing constructions of elusive groups mostly arise from split extensions. In this paper, we initiate the construction of elusive groups via non-split extensions. As a demonstration, we construct elusive groups of new degrees, namely for each Mersenne prime and integer . We also construct the first examples of elusive groups with odd degree, namely , and twice odd degree, namely for each . We conclude by proposing further problems to advance this new direction of research.

Paper Structure

This paper contains 5 sections, 13 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Illustration of idea
  3. Elusive groups of degree $p^{3k-4}(p+1)/2$
  4. Elusive groups of degree not divisible by $4$
  5. Concluding remarks

Key Result

Theorem 1.3

For each Mersenne prime $p\geq7$ and integer $k\geq2$, there exists an elusive group of degree $p^{3k-4}(p+1)/2$ which has the form and stabilizer $\mathrm{C}_p^2{:}\mathrm{D}_{p-1}$.

Theorems & Definitions (32)

  • Conjecture 1.1: Polycirculant Conjecture
  • Theorem 1.3
  • Remark
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Example 2.2
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • ...and 22 more