2-Groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs
Du, Shao Fei ...
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.
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