On groups with a given central factor

TL;DR

This paper classifies certain groups based on their central factors and centralizers, providing new insights into their structure and extending previous classifications for groups with specific central factors.

Contribution

It introduces a classification of groups with a given central factor and groups with limited centralizers, improving upon earlier results in group theory.

Findings

Classified groups with a central factor of order p^3

Classified groups with at most nine element centralizers

Extended previous classifications in group theory

Abstract

We have classified, upto isoclinism, certain groups with a given central factor. As an application, we classify, upto isoclinism, groups having at the most nine element centralizers. Among other results of independent interest, we have classified, upto isoclinism, groups having a central factor of order $p^3$, $p$ a prime. All these improves some previous results.