Abelianization of p-groups with derived subgroup of prime order

TL;DR

This paper investigates the structure of the abelianization of p-groups with a focus on the derived subgroup of prime order, providing new insights into their algebraic properties.

Contribution

It offers a detailed description of the abelianization structure derived from group presentations and clarifies the nature of the Frattini subgroup in these groups.

Findings

Derived the structure of G^{ab} from presentations.

Proved the Frattini subgroup is G^p or G^p/G'.

Clarified the relationship between G', G^p, and the Frattini subgroup.

Abstract

In this paper, we arrive at the structure for $G^{ab}(= G/G')$ 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 $G'$.