On finite capable $p$-groups of class 2 with cyclic commutator subgroups

TL;DR

This paper investigates finite capable p-groups of class 2 with cyclic commutator subgroups, focusing on their structural properties and conditions for capability within group theory.

Contribution

It characterizes finite capable p-groups of class 2 with cyclic commutator subgroups where the center is contained in the Frattini subgroup, advancing understanding of their structure.

Findings

Identifies conditions for capability in these groups

Provides structural classifications for such p-groups

Enhances understanding of the relationship between center and Frattini subgroup

Abstract

We study finite capable $p$-groups $G$ of nilpotency class 2 such that the commutator subgroup $\gamma_2(G)$ of $G$ is cyclic and the center of $G$ is contained in the Frattini subgroup of $G$.