Let $G$ be a finite group. I am interested in finite groups having a non-trivial subgroup $H$ $(H\neq \{e\}, G$) such that $H' \subseteq H$ or $H \subseteq H'$ for every subgroup $H'\neq H.$ That is, there is some non-trivial subgroup $H$ which contains or it is contained in the rest of subgroups.
For instance, if we assume that $G$ is finite and cyclic, this condition is satisfied only if $G \cong \mathbb{Z}_{p^k}$ for $p$ prime. If $G \cong \mathbb{Z}_{p^k},$ every non-trivial subgroup $H$ contains or it is contained in the rest of subgroups. In other words, in this particular case, the subgroup lattice of $G$ is a chain. However, for me it is enough if exists some $H$ like this.
I wonder if it is possible to derive some result characterizing groups having a subgroup $H$ as above in the general case of finite groups (or at least abelian ones). Any help is very welcome.
Thank you for your time in advance!
Best Answer
Suppose that $G$ is finite, and let $H$ be a nontrivial proper subgroup such that $H' \leq H$ or $H \leq H'$ for all subgroups $H' \leq G$.
I will leave the details to you, but here is a start:
In the end you should be able to see that $G$ is a $p$-group with a unique subgroup of order $p$. By a classical theorem, such a group is cyclic or generalized quaternion.