There is one group of genus two
type: Item
instance of: scholarly article
title: There is one group of genus two
zbMATH Open document ID: 0537.05022
DOI: 10.1016/0095-8956(84)90032-7
author: Thomas W. Tucker
published in: Journal of Combinatorial Theory. Series B
publication date: 1984-01-01T00:00:00Z
review text: The genus of a finite group \(\Gamma\) is the minimum genus over all closed orientable 2-manifolds in which a Cayley graph for \(\Gamma\) can be imbedded; that is, it is the minimum genus of all Cayley graphs \(G_{\Delta}(\Gamma)\) for \(\Gamma\), the minimum being taken over all possible generating sets \(\Delta\) for \(\Gamma\). There are infinitely many groups of genus zero (the cyclic groups \(Z_ n\), for example) and of genus one \((Z_ n\times Z_ n\), when \(n\geq 3\), for example), but only finitely many groups of a given genus two or more [\textit{T. W. Tucker}, Trans. Am. Math. Soc. 258, 167-179 (1980; Zbl 0444.05039)]. In the present paper, the same author shows that there is exactly one group of genus two: \(\Gamma =<x,y,z| \quad x^ 2=y^ 2=z^ 2=(xy)^ 2=(yz)^ 3=(xz)^ 8=(y(xz)^ 4)^ 2=1>; | \Gamma | =96\).
Mathematics Subject Classification ID: 05C10
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 3853100
zbMATH Keywords: genus of a finite group
zbMATH Keywords: Cayley graph
reviewed by: Arthur T. White
MaRDI profile type: Publication
full work available at URL: 0095-8956(84)90032-7
OpenAlex ID: W2034691552
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description: scientific article; zbMATH DE number 3853100