Strong digraph groups

A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each suc...

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Bibliographic Details
Published in:Canadian mathematical bulletin Vol. 67; no. 4; pp. 991 - 1000
Main Authors: Cihan, Mehmet Sefa, Williams, Gerald
Format: Journal Article
Language:English
Published: Canada Canadian Mathematical Society 01.12.2024
Cambridge University Press
Subjects:
Apexes
Graph theory
Digraph group
digraph products
05C20
Cayley digraph
05C40
20F05
05C25
05C38
strong digraph
circulant digraph
ISSN:0008-4395, 1496-4287
Online Access:Get full text
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Summary:A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$ , where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.
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ISSN:0008-4395
1496-4287
DOI:10.4153/S0008439524000390