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 d(G) of generators of a finite Alperin p-group G is n a parts per thousand yen 3, then d(G') a parts per thousand currency sign C (n) (2) for p not equal 3 and d(G') a parts per thousand currency sign C (n) (2) + C (n) (3) for p = 3. The first section of the paper deals with finite Alperin p-groups G with p not equal 3 and d(G) = n a parts per thousand yen 3 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 homocyclic commutator subgroup G- of rank C (n) (2) + C (n) (3) , then GaEuro(3) is an elementary abelian group.