A class of nonabelian nonmetacyclic finite 2-groups

Abstract

Abstract. Nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic and possessing at least one nonabelian and at least one nonmetacyclic proper subgroup have been investigated and classified. Using the obtained result and two previously known results one gets the complete classification of all nonabelian nonmetacyclic finite 2-groups in which every proper subgroup is abelian or metacyclic.

The classification of finite groups by using iteration digraphs

“…Then it can be further classified into two categories: in the first all proper subgroups of G are metacyclic and in the other G has at least one proper non-metacyclic subgroup. In either case, by using Theorem 3.2 of [5] and the main theorem of [6], G must be generated by three generators, which contradicts our assumption.…”