Research Article
Year 2025,
, 410 - 423, 30.06.2025
Abstract
References
- [1] Cuno J., Williams G., A class of digraph groups defined by balanced presentations, Journal of Pure and Applied Algebra., 224(8) (2020) 106342.
- [2] Cihan M.S., Williams G., Finite groups defined by presentations in which each defining relator involves exactly two generators, Journal of Pure and Applied Algebra 228 (4) (2024) 107499.
- [3] Johnson D.L., Topics in the theory of group presentations, London Mathematical Society Lecture Note Series, 42. Cambridge University Press, (1980).
- [4] Johnson D.L., Robertson E.F., Finite groups of deficiency zero, In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, (36) 1979 275-289.
- [5] Cihan M.S., Digraph groups corresponding to digraphs with one more vertex than arcs, European Journal of Science and Technology., (41) (2022) 31–35.
- [6] Pride S.J., Groups with presentations in which each defining relator involves exactly two generators, J. Lond. Math. Soc., II. Ser. 36 (1-2) (1987) 245–256.
- [7] Bogley W.A., Williams G., Efficient finite groups arising in the study of relative asphericity, Math. Z. 284(1) (2016) 507–535.
- [8] Cihan M.S., Digraph Groups and Related Groups, Doctoral dissertation, University of Essex, 2022.
Year 2025,
, 410 - 423, 30.06.2025
Abstract
This paper investigates a particular class of digraph groups that are defined by non-empty balanced presentations. Each relation is expressed in the form R(x,y), where x and y are distinct generators, and R(⋅,⋅) is based on a fixed cyclically reduced word R(a,b) involving both a and b. A directed graph is constructed for each such presentation, where vertices correspond to generators and edges represent the relations. In previous research, Cihan identified 35 families of digraphs that satisfy |V(Γ)|=|A(Γ)|-1, of which 11 of them do not contain leaves. This paper demonstrates that, with two exceptions, the rank of the associated groups is either 1 or 2.
References
- [1] Cuno J., Williams G., A class of digraph groups defined by balanced presentations, Journal of Pure and Applied Algebra., 224(8) (2020) 106342.
- [2] Cihan M.S., Williams G., Finite groups defined by presentations in which each defining relator involves exactly two generators, Journal of Pure and Applied Algebra 228 (4) (2024) 107499.
- [3] Johnson D.L., Topics in the theory of group presentations, London Mathematical Society Lecture Note Series, 42. Cambridge University Press, (1980).
- [4] Johnson D.L., Robertson E.F., Finite groups of deficiency zero, In Homological group theory (Proc. Sympos., Durham, 1977), London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge-New York, (36) 1979 275-289.
- [5] Cihan M.S., Digraph groups corresponding to digraphs with one more vertex than arcs, European Journal of Science and Technology., (41) (2022) 31–35.
- [6] Pride S.J., Groups with presentations in which each defining relator involves exactly two generators, J. Lond. Math. Soc., II. Ser. 36 (1-2) (1987) 245–256.
- [7] Bogley W.A., Williams G., Efficient finite groups arising in the study of relative asphericity, Math. Z. 284(1) (2016) 507–535.
- [8] Cihan M.S., Digraph Groups and Related Groups, Doctoral dissertation, University of Essex, 2022.
There are 8 citations in total.
Details
| Primary Language | English |
|---|---|
| Subjects | Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics) |
| Journal Section | Natural Sciences |
| Authors | |
| Publication Date | June 30, 2025 |
| Submission Date | March 12, 2025 |
| Acceptance Date | June 11, 2025 |
| Published in Issue | Year 2025 |