Commutators in groups of order $p^7$

TL;DR

This paper characterizes groups of order p^7 where not all elements of the commutator subgroup are commutators, providing new structural insights into such p-groups.

Contribution

It offers a novel characterization of p^7 groups with non-fully generated commutator subgroups and derives structural results for these groups.

Findings

Identification of conditions where not all commutator subgroup elements are commutators

Structural properties of groups of order p^7

Characterization of specific p-group structures

Abstract

We present a characterisation of groups $G$ of order $p^7$, $p$ prime, in which not all elements of the commutator subgroup $\gamma_2(G)$ of $G$ are commutators in $G$. On the way we obtain several structural results on groups of order $p^7$.