2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs - Du, Shao Fei (Ars mathematica contemporanea, 2013)
We classify those 2-groups ?$G$? which factorise as a product of two disjoint cyclic subgroups ?$A$? and ?$B$?, transposed by an automorphism of order 2. The case where ?$G$? is metacyclic having been dealt with elsewhere, we show that for each ?$e \ge 3$? there are exactly three such non-metacyclic groups ?$G$? with ?$|A| = |B| = 2^e$?, and for ?$e = 2$? there is one. These groups appear in a classification by Berkovich and Janko of 2-groups with one non-metacyclic maximal subgroup; we enumerate these groups, give simpler presentations for them, and determine their automorphism groups.
Keywords: complete bipartite graph; product of cyclic groups; regular map; teorija grafov