Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups

Details Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups Transitive Group Actions: (IM)PRIMITIVITY and Semiregular Subgroups

2007-01-24

arxiv.org/abs/math/0701686v1

Mathematics

Group Theory

26 pages, 0 figures

Scientific paper

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 $X$ and give their description in terms of orbits of $H$. Conditions and descriptions presented in this paper are derived by studying spectral properties of the associated $G$-invariant digraphs arising as regular abelian covers. As an essential tool, irreducible complex characters defined on $H$, are used. Questions of this kind arise naturally when classifying combinatorial objects which enjoy a certain degree of symmetry. The techniques of this paper allow one to consider such problems without referring to, as it often is the case, the classification of finite simple groups. As an illustration, a new and short proof of an old result of Frucht, Graver and Watkins ({\it Proc. Camb. Phil. Soc.}, {\bf 70} (1971), 211-218) classifying edge-transitive generalized Petersen graphs, is given.

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