Circulant Homogeneous Factorisations of Complete Digraphs Kpd with p an Odd Prime

Let F = (Kn,P) be a circulant homogeneous factorisation of index k, that means P is a partition of the arc set of the complete digraph Kn into k circulant factor digraphs such that there exists σ ∈ Sn permuting the factor circulants transitively amongst themselves. Suppose further such an element σ normalises the cyclic regular automorphism group of these circulant factor digraphs, we say F is normal. Let F = (Kpd ,P) be a circulant homogeneous factorisation of index k where pd, (d > 1) is an odd prime power. It is shown in this paper that either F is normal or F is a lexicographic product of two smaller circulant homogeneous factorisations.