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(·, ·) is determined by some fixed cyclically reduced word R(a, b) that involves both a and b Associated to each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this paperwe consider digraph groups for strong digraphs that are digon-free and trianglefree. 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.