Capability of nilpotent products of cyclic groups

TL;DR

This paper investigates the capability of nilpotent products of cyclic groups, generalizing Baer's theorem for small class cases and providing new necessary conditions for p-groups of arbitrary class.

Contribution

It extends existing results on group capability to broader classes of nilpotent groups and introduces new necessary conditions for p-groups of any class.

Findings

Generalization of Baer's theorem for small class nilpotent groups

Necessary condition for capability of p-groups of class k

Results on capability of certain nilpotent groups of class 2

Abstract

A group is called capable if it is a central factor group. We consider the capability of nilpotent products of cyclic groups, and obtain a generalization of a theorem of Baer for the metabelian small class case. The approach is also used to obtain some recent results on the capability of certain nilpotent groups of class 2. We also prove a necessary condition for the capability of an arbitrary p-group of class k, and some further results.