  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 Frucht, Graver and Watkins ({\it Proc. Camb. Phil.
Soc.}, {\bf 70} (1971), 211-218) classifying edge-transitive generalized
Petersen graphs, is given.
