Elusive groups from non-split extensions

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 3 k − 4 ( p + 1 ) / 2 for each Mersenne prime p ≥ 7 and integer k ≥ 2 . We also construct the first examples of elusive groups with odd degree, namely 3 k + 1 ⋅ 5 2 , and twice odd degree, namely 2 ⋅ 3 k + 1 ⋅ 5 2 for each k ≥ 1 . We conclude by proposing further problems to advance this new direction of research.