Finite 2-groups with exactly one nonmetacyclic maximal subgroup.
It is well-known that a \(p\)-group which is the product of two finite cyclic groups with \(p\) an odd prime is necessarily metacyclic, and such nonmetacyclic 2-group contains exactly one nonmetacyclic maximal subgroup. The aim of the present paper is to characterize all finite 2-groups \(G\) with exactly one nonmetacyclic maximal subgroup. This turns out to be an extremely complicated task. The group \(G\) is generated by two or three elements, in the former case the group \(G\) is the product of two cyclic subgroups, in the latter case the commutator subgroup is elementary Abelian of order at most 4. In the paper all these groups are determined in terms of generators and relations and their structures are described.
scientific article; zbMATH DE number 5351786