2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs
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 ≥ 3 there are exactly three such non-metacyclic groups G with ∣A∣ = ∣B∣ = 2e, 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.
2-groups, non-metacyclic, complete bipartite graphs