Abelianization of p-groups with derived subgroup of prime order

In this paper, we arrive at the structure for $G^{ab}(=G\mathrm{/}{G}^{\mathrm{\prime }})$ from a presentation for $G$. We also prove that the frattini subgroup of the abelianization is $G^p$ or the quotient of $G^p$ by the derived subgroup $G^{\mathrm{\prime }}$.