On a class of highly symmetric \(k\)-factorizations (Q1953405)

Summary: A \(k\)-factorization of \(K_v\) of type \((r, s)\) consists of \(k\)-factors each of which is the disjoint union of \(r\) copies of \(K_{k+1}\) and \(s\) copies of \(K_{k,k}\). By means of what we call the patterned \(k\)-factorization \(F_k(D)\) over an arbitrary group \(D\) of order \(2s + 1\), it is shown that a \(k\)-factorization of type \((1, s)\) exists for any \(k\geq2\) and for any \(s\geq1\) with \(D\) being an automorphism group acting sharply transitively on the factor-set. The general method to construct a \(k\)-factorization \(F\) of type \((1, s)\) over an arbitrary 1-factorization \(S\) of \(K_{2s+2} (F\) is said to be based on \(S\)) is used to prove that the number of pairwise non-isomorphic \(k\)-factorizations of this type goes to infinity with \(s\). In this paper, we show that the full automorphism group of \(F\) is known as soon as we know the one of \(S\). In particular, the full automorphism group of \(F_k(D)\) is determined for any \(k\geq2\), generalizing a result given by \textit{P. J. Cameron} for patterned 1-factorizations [J. Lond. Math. Soc., II. Ser. 11, 337--346 (1975; Zbl 0312.05107)]. Finally, it is shown that \(F_k(D)\) has exactly \((k!)^{2s+1}(2s+1)|Aut(D)|\) automorphisms whenever \(D\) is abelian.