Commutators and commutator subgroups of finite $p$-groups

TL;DR

This paper classifies finite p-groups with a specific commutator subgroup of order p^4 and exponent p, focusing on cases where not all elements are commutators, advancing understanding of their structure.

Contribution

It provides a detailed classification of finite p-groups with a particular commutator subgroup structure, highlighting cases where not all elements are commutators.

Findings

Classification of p-groups with gamma_2(G) of order p^4 and exponent p

Identification of cases where not all elements of gamma_2(G) are commutators

Structural insights into finite p-groups based on their commutator subgroups

Abstract

We present a classification of finite $p$-groups $G$ with $\gamma_2(G)$, the commutator subgroup of $G$, of order $p^4$ and exponent $p$ such that not all elements of $\gamma_2(G)$ are commutators.