Embedding groups of class two and prime exponent in capable and non-capable groups

TL;DR

This paper investigates the embedding of certain p-groups into larger groups of class two and exponent p, demonstrating the existence of both capable and non-capable embeddings with specific structural bounds.

Contribution

It introduces a method to embed p-groups of class two and exponent p into larger groups that are either capable or non-capable, with explicit bounds on their abelianization ranks.

Findings

Existence of embeddings into capable and non-capable groups

Upper bounds for ranks of abelianizations of the embedding groups

Embedding groups cannot always be expressed as a central product

Abstract

We show that if $G$ is any $p$-group of class at most two and exponent $p$, then there exist groups $G_1$ and $G_2$ of class two and exponent $p$ that contain $G$, neither of which can be expressed as a central product, and with $G_1$ capable and $G_2$ not capable. We provide upper bounds for ${\rm rank}(G_i^{\rm ab})$ in terms of ${\rm rank}(G^{\rm ab})$ in each case.