On finite Alperin $p$-groups with homocyclic commutator subgroup

We study metabelian Alperin groups, i.e., metabelian groups in which every 2-generated subgroup has a cyclic commutator subgroup. It is known that, if the minimum number of generators $d(G)$ of a finite Alperin $p$-group $G$ is $n\geq3$, then $d(G')\leq C_n^2$ for $p\neq3$ and $d(G')\leq C_n^2+C_n^3$ for $p=3$. The first section of the paper deals with finite Alperin $p$-groups $G$ with $d(G)\geq3$ and $p\neq3$ that have a homocyclic commutator subgroup of rank $C_n^2$. In addition, a corollary is deduced for infinite Alperin $p$-groups. In the second section, we prove that, if $G$ is a finite Alperin $3$-group with a homocyclic commutator subgroup $G'$ of rank $C_n^2+C_n^3$, then $G'$ is an elementary abelian group.