Strong digraph groups

Cihan, Mehmet Sefa  ; Williams, Gerald 

Canadian mathematical bulletin, Tome 67 (2024) no. 4, pp. 991-1000

Résumé

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.

Détail

DOI : 10.4153/S0008439524000390

Mots-clés : Digraph group, strong digraph, Cayley digraph, circulant digraph, digraph products