Finite nonabelian 2-groups all of whose minimal nonabelian subgroups are metacyclic and have exponent 4

Abstract

We classify here the title groups and note that such groups must be of exponent >4 if both D 8 and H 2 = 〈 a , b | a 4 = b 4 = 1 , a b = a − 1 〉 appear as subgroups (Theorem 1.1). This solves a problem stated by Y. Berkovich in [Y. Berkovich, Groups of Prime Power Order, I and II (with Z. Janko), Walter de Gruyter, Berlin, 2008]. On the other hand, if G is a nonabelian finite 2-group all of whose minimal nonabelian subgroups are non-metacyclic and have exponent 4, then G must be of exponent 4 (Theorem 1.5). We also solve a more general problem Nr. 1475 of Berkovich [Y. Berkovich, Groups of Prime Power Order, I and II (with Z. Janko), Walter de Gruyter, Berlin, 2008] by classifying nonabelian finite 2-groups of exponent 2 e ( e ⩾ 3 ) which do not have any minimal nonabelian subgroup of exponent 2 e (Theorem 1.6). Finally, we prove Lemma 1.7 which might be useful for future investigations.