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

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 $\mathrm{rank}(G_i^{\mathrm{ab}})$ in terms of $\mathrm{rank}(G^{\mathrm{ab}})$ in each case.

5 pages; title change, minor corrections

References in corpus (1)

• On the capability of finite groups of class two and prime exponent