Document Type : Original Manuscript


Department of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili, P.O.Box 56199-11367, Ardabil, Iran.



The infinite family of groups defined by the presentation $G_p=\langle x, y|x^p=y^p,\; xyx^my^n=1\rangle$, in which $p$ is a prime in $\{2,3,5\}$ and $m,n\in\mathbb{N}_0$, will be considered and finite and infinite groups in the family will be determined. For the primes $p=2,3$ the group $G_p$ is finite and for $p=5$, the group is finite if and only if $m\equiv n\equiv1\pmod5$ is not the case.


1. H. Abdolzadeh and R. Sabzchi, An infinite family of finite 2-groups with deficiency zero, Int. J. Group Theory, Vol. 6(3) (2017), 45–49.

2. M. J. Beetham and C. M. Campbell, A note on the Todd-Coxeter coset enumeration algorithm, P. Edinburgh Math. Soc. 20 (1976) 73–79.

3. G. Havas, M. F. Newman and E. A. O’Brien, Groups of deficiency zero, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 25 (1994) 53–67.

4. D. L. Johnson, Topics in the theory of group presentations, London Math. Soc. Lecture Note Ser., 42 Cambridge University Press, Cambridge, 1980.

5. R. Sabzchi and H. Abdolzadeh, An infinite family of finite 3-groups with deficiency zero, J. Algebra Appl., Vol. 18(7) (2019), 1–12.

6. The GAP Group, GAP | Groups, Algorithms and Programming, Version 4.4 (available from, 2005.