On finite Alperin 2-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today??s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z ₂ and Z ₄ and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.