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

Shaofei Du, Gareth Jones, Jin Ho Kwak, Roman Nedela, Martin Škoviera

Abstract


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.

Keywords


Regular map, complete bipartite graph, product of cyclic groups.

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DOI: https://doi.org/10.26493/1855-3974.295.270

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications