Finite nonabelian 2-groups in which any two noncommuting elements generate a subgroup of maximal class

摘要

We determine here the structure of the title groups. It turns out that such a group G is either quasidihedral or G = HZ(G), where H is of maximal class or extraspecial and (Z(G)) ≤ Z(H). This solves a problem stated by Berkovich. The corresponding problem for p > 2 is open but very difficult since the p-groups of maximal class are not classified for p > 2.