Shahrood University of TechnologyJournal of Algebraic Systems2345-51289120210901DEFICIENCY ZERO GROUPS IN WHICH PRIME POWER OF GENERATORS ARE CENTRAL3543205110.22044/jas.2020.9361.1456ENM. AhmadpourDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
P.O.Box 56199-11367, Ardabil, Iran.H. AbdolzadehDepartment of Mathematics, Faculty of Sciences, University of Mohaghegh Ardabili,
P.O.Box 56199-11367, Ardabil, Iran.Journal Article20200203The 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.https://jas.shahroodut.ac.ir/article_2051_e39e91ea25ff14b11a5e14de63c7d0a3.pdf