Regular embeddings of K n , n where n is a power of 2 . II : Non - metacyclic case

In this paper, a classification of the regular embeddings of Kn,n, where n = 2e is a power of two, is completed. The number of such regular maps is one or two for e = 1, 2, respectively. For e ≥ 3 there are 2e−2 + 4 regular embeddings of K2e,2e . The method is based on classification of groups G which factorise as a product of two cyclic groups G = XY of order n such that the cyclic factors are transposed by an involutory automorphism. The case of metacyclic G is analysed in the preceding paper. In the present paper we show that for each exponent e ≥ 3 there are exactly three non-metacyclic groups G = XY with |X| = |Y | = 2e. Given n = 2e the three non-metacyclic groups G give rise to four regular regular embeddings of Kn,n.