Year: 2025
Type: article
Abstract: A classical theorem of Jordan asserts that if a group G acts transitively on a finite set of size at least 2 $2$ 2 , then G contains a derangement (a fixed-point free element). Generalisations of Jordan’s theorem have been studied extensively, due in part to their applications in graph theory, number theory and topology. We address a generalisation conjectured recently by Ellis and Harper [‘Orbits of permutation groups with no derangements’, Preprint, 2024, arXiv:2408.16064], which says that if G has exactly two orbits and those orbits have equal length n greater than or equals 2 $n \geq 2$ n ≥ 2 , then G contains a derangement. We prove this conjecture in the case where n is a product of two primes, and in the case where n equals b p $n=bp$ n = b p with p a prime and 2 b less than or equals p $2b\leq p$ 2 b ≤ p . We also verify the conjecture computationally for n less than or equals 30 $n \leq 30$ n ≤ 30 .
Authors: MELISSA LEE, TOMASZ POPIEL, GABRIEL VERRET
Institutions: Monash University, University of Auckland
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $3,550
Funder: Australian Research Council
OpenAlex ID: W4417025502