Let $P$ be a transitive permutation group of order $p^m$, $p$ an odd prime, containing a regular cyclic subgroup. The main result of this paper is a determination of the suborbits of $P$. The main result is used to give a simple proof of a recent result by J.~Morris on Cayley digraph isomorphisms.

MSC Classifications:

20B25, 05C60 show english descriptions Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX]
Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.) 20B25 - Finite automorphism groups of algebraic, geometric, or combinatorial structures [See also 05Bxx, 12F10, 20G40, 20H30, 51-XX]
05C60 - Isomorphism problems (reconstruction conjecture, etc.) and homomorphisms (subgraph embedding, etc.)

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