Elusive groups from non-split extensions

TL;DR

This paper introduces new constructions of elusive groups via non-split extensions, expanding known degrees and types of elusive groups, and relates to the Polycirculant Conjecture in algebraic graph theory.

Contribution

It pioneers the construction of elusive groups through non-split extensions, including new degrees and odd degree examples, advancing understanding of elusive groups in permutation group theory.

Findings

Constructed elusive groups of degrees involving Mersenne primes

First examples of elusive groups with odd degree

First examples of elusive groups with twice odd degree

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.