About non-normal nilpotent subgroups

Question

Suppose that all non-normal abelian subgroups of a finite group $G$ are cyclic. What can I say about non-normal nilpotent subgroups of $G$? Is it true that such supgroups are cyclic?

I appreciate your help.

Answer 1

The finite p-groups all of whose non-normal abelian subgroups are cyclic have been classified -see here. I would think that among these $p$-groups we should find a counterexample.

What about the generalized quaternion group $Q_{32}$, where we know that all abelian subgroups are cyclic, and the nilpotent subgroup $Q_8$ is not normal and not cyclic? See here.

Answer 2

If All non-normal nilpotent subgroups of a group G are cyclic, then all non-normal abelian subgroups of G are cyclic. But the converse is not true. As an example, generalized Quaternion group of orders greater than 16. The Quaternion type groups contain two non-normal cyclic subgroups of order 4, and other non-normal subgroups are non-abelian of the Quaternion type.

The structure of finite groups whose non-normal nilpotent subgroups are cyclic are characterized and can see in the following paper.

H. Mousavi and G. Tiemouri, Structure of finite groups with trait of non-normal subgroups, Communications in Algebra, Vol. 47, No. 3, (2019).

The structure of finite groups whose non-normal abelian subgroups are cyclic are characterized and can see in the following conference paper.

English Booklet (Vol 1) of 51st Annual Iranian Mathematics Conference, (119-122).