Finitely Generated Groups $G$ such that $G/Z(G) \simeq C_p \times C_p$

TL;DR

This paper classifies finitely generated groups G where the quotient by the center is isomorphic to a product of two cyclic groups of prime order p, extending previous results from the case p=2.

Contribution

It generalizes the classification of groups with a specific quotient structure from p=2 to arbitrary prime p, providing explicit descriptions.

Findings

Explicit classification of finitely generated groups with G/Z(G) ≅ C_p × C_p.

Extension of previous results from p=2 to general prime p.

Provides structural descriptions of these groups.

Abstract

Finite groups $G$ such that $G/Z(G) \simeq C_2 \times C_2$ where $C_2$ denotes a cyclic group of order 2 and $Z(G)$ is the center of $G$ were studied in \cite{casofinito} and were used to classify finite loops with alternative loop algebras. In this paper we extend this result to finitely generated groups such that $G/Z(G) \simeq C_p \times C_p$ where $C_p$ denotes a cyclic group of prime order $p$ and provide an explicit description of all such groups.