Geodesic
Transitive group actions: (im)primitivity and semiregular subgroups.
Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 867-885.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

The following problem is considered: if $H$ is a semiregular abelian subgroup of a transitive permutation group $G$ acting on a finite set $X$, find conditions for (non)existence of $G$-invariant partitions of $X$. Conditions presented in this paper are derived by studying spectral properties of associated $G$-invariant digraphs. As an essential tool, irreducible complex characters of $H$ 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 R. Frucht, J. E. Graver and M. E. Watkins [Proc. Camb. Philos. Soc. 70, 211-218 (1971; Zbl 0221.05069)] classifying edge-transitive generalized Petersen graphs, is given.
Classification : 20B20, 20B05, 05C25, 05E18
Keywords: transitive permutation groups, semiregular subgroups, transitive group actions, primitive permutation groups, imprimitivity, digraphs, generalized Petersen graphs 
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     title = {Transitive group actions: (im)primitivity and semiregular subgroups.},
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Kovács, István; Malnič, Aleksander; Marušič, Dragan; Miklavič, Štefko. Transitive group actions: (im)primitivity and semiregular subgroups.. Journal of Algebraic Combinatorics, Tome 41 (2015) no. 3, pp. 867-885. http://geodesic.mathdoc.fr/item/JAC_2015__41_3_a2/