DEFICIENCY ZERO GROUPS IN WHICH PRIME POWER OF GENERATORS ARE CENTRAL

Abstract

The infinite family of groups defined by the presentation $G_p=langle x, y|x^p=y^p,; xyx^my^n=1rangle$, in which $p$ is a prime in ${2,3,5}$ and $m,ninmathbb{N}_0$, 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 $mequiv nequiv1pmod5$ is not the case.