Digraph groups and related groups

Digraph groups and related groups

Groups can be expressed in terms of a finite a digraph which vertices correspond to the generators and arcs correspond to the relators. Cuno and Williams investigated when the number of vertices is equal the number of arcs, where the undirected graph is triangle free that means the girth is at least 4, and they proved that the corresponding group is either finite cyclic or infinite. It is known that when the number of vertices is more than the number of arcs, then it is infinite.

Therefore, I investigated when the number of vertices is less than or equal the number of arcs in my thesis. But it is more interesting when the undirected graph is with triangle and therefore the underlying graph is complete graph. When we directed the complete graph, then it is known as tournaments.

All known examples are done by Mennicke and Johnson for a strong tournament with 3 vertices. In 1959, Mennicke provided an example of a group defined by the presentation M(a, b, c) =〈x, y, z | y^−1xy=x^a, z^−1yz=y^b, x^−1zx=z^c, which is finite in the case a=b=c ≥ 3. In 1997,  Johnson provided another group needing exactly three generators with presentation J(a, b, c) =〈x, y, z|x^y=y^(b−2)x^−1y^(b+2), y^z=z^(c−2)y^−1z^(c+2),z^x=x^(a-2)z^−1x^(a+2) and which is finite in the cases where a, b,  and c are non-zero even integers. These are important since they provide examples of finite groups needing exactly three generators.

In this talk, I will talk about generalisation of their groups from 3 generators to n generators for all strong tournaments.

Speaker

Mehmet Ciha, University of Essex

How to attend

If not a member of the Dept. Mathematical Science at the University of Essex, you can register your interest in attending the seminar and request the Zoom’s meeting password by emailing Dr Jesus Martinez-Garcia (jesus.martinez-garcia@essex.ac.uk)