2-groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1309.20010
DOI: 10.26493/1855-3974.295.270
author: Gareth Jones
author: Jin Ho Kwak
author: Roman Nedela
author: Martin Škoviera
author: Shaofei Du
published in: Ars Mathematica Contemporanea
publication date: 2014-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/1112.2376
Mathematics Subject Classification ID: 20D15
Mathematics Subject Classification ID: 20D40
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 6359792
zbMATH Keywords: finite 2-groups
zbMATH Keywords: regular maps
zbMATH Keywords: complete bipartite graphs
zbMATH Keywords: products of cyclic groups
MaRDI profile type: Publication
OpenAlex ID: W2168229784
arXiv ID: 1112.2376
Wikidata QID: Q129355438
title: 2-groups that factorise as products of cyclic groups, and regular embeddings of complete bipartite graphs.
review text: 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.
reviewed by: János Kurdics
description: scientific article; zbMATH DE number 6359792