2-groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs.
Let \(G\) be a nonabelian finite group. \(G\) is called \(n\)-isobicyclic if it is the product of two cyclic groups of order \(n\) with trivial intersection and with an automorphism transposing two generators of those cyclic groups. These groups have arisen recently in the theory of \(p\)-groups of \textit{Y. Berkovich} and \textit{Z. Janko} [Groups of prime power order. Vol. 2. Berlin: Walter de Gruyter (2008; Zbl 1168.20002)], and concerning embeddings of the complete bipartite graphs on orientable surfaces, a recent paper is by \textit{G. A. Jones} [Proc. Lond. Math. Soc. (3) 101, No. 2, 427-453 (2010; Zbl 1202.05028)].NEWLINENEWLINE Description of \(n\)-isobicyclic groups may be reduced to the prime-power order case, and for odd primes these groups are metacyclic and their description is well-known. The remaining non-metacyclic case with \(n=2^e\) (\(e\geq 2\)) is treated in the present paper, there is one family of three parameters of these groups described in terms of generators and relations. Moreover, applying this result, the authors determine the automorphism groups of these groups and reprove a theorem on embedding of the complete bipartite graph.
scientific article; zbMATH DE number 6359792