Circulant homogeneous factorisations of complete digraphs \(\mathbf K_{p^{d}}\) with \(p\) an odd prime (Q2628257)

Summary: Let \(\mathcal F=(\mathbf K_{n},\mathcal P)\) be a circulant homogeneous factorisation of index \(k\), that means \(\mathcal P\) is a partition of the arc set of the complete digraph \(\mathbf K_n\) into \(k\) circulant factor digraphs such that there exists \(\sigma\in S_n\) permuting the factor circulants transitively amongst themselves. Suppose further such an element \(\sigma\) normalises the cyclic regular automorphism group of these circulant factor digraphs, we say \(\mathcal F\) is normal. Let \(\mathcal F=(\mathbf K_{p^{d}},\mathcal P)\) be a circulant homogeneous factorisation of index \(k\) where \(p^d\), (\(d\geq 1\)) is an odd prime power. It is shown in this paper that either \(\mathcal F\) is normal or \(\mathcal F\) is a lexicographic product of two smaller circulant homogeneous factorisations.