Transitive group actions: (im)primitivity and semiregular subgroups

Abstract

The following problem is considered: if $$H$$H is a semiregular abelian subgroup of a transitive permutation group $$G$$G acting on a finite set $$X$$X, find conditions for (non)existence of $$G$$G-invariant partitions of $$X$$X. Conditions presented in this paper are derived by studying spectral properties of associated $$G$$G-invariant digraphs. As an essential tool, irreducible complex characters of $$H$$H are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. As an illustration, a new and short proof of an old result of Frucht et al. (Proc Camb Philos Soc 70:211–218, 1971) classifying edge-transitive generalized Petersen graphs, is given.