On a Class of Highly Symmetric k-Factorizations

A k–factorization of Kv of type (r, s) consists of k–factors each of which is the disjoint union of r copies of Kk+1 and s copies of Kk,k. By means of what we call the patterned k–factorization Fk(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 > 2 and for any s > 1 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 K2s+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 Fk(D) is determined for any k > 2, generalizing a result given by P. J. Cameron for patterned 1–factorizations [J London Math Soc 11 (1975), 189–201]. Finally, it is shown that Fk(D) has exactly (k!)2s+1(2s + 1)|Aut(D)| automorphisms whenever D is abelian.