Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial

Question

The group of rational numbers $(\mathbb Q,+)$ has an interesting property , that the intersection of any two non-trivial subgroups of this group is non-trivial . Let us call this property the " non-trvial intersection property " or NIP in short . Now it is easy to see that this NIP property is invariant under group isomorphism , so if $G$ is a group having NIP , then $G$ cannot be isomorphic with $H \times K$ (because if $|H|,|K|>1$ , then $\{e_H\} \times K , H\times \{e_K\}$ are non-trivial subgroups of $H \times K$ with trivial intersection ) for any groups $H$ and $K$ .

I am looking for more examples of groups having NIP , does there exist infinitely many non-isomorphic such groups ? Also , have this kind of groups been studied ? Any reference or link will also be very helpful . Thanks in advance

NOTE : All groups considered are to be meant with more than one element

Answer 1

The groups ${\mathbb Z}_{p^{\infty}}$ with $p$ prime have this property, and there infinitely many of these - one for each prime. In fact they have a unique minimal nontrivial subgroup, which is cyclic of order $p$. Although the groups are infinite, all of their proper subgroups are finite cyclic $p$-groups.

You could define ${\mathbb Z}_{p^{\infty}}$ to be the multiplicative group of all complex $p^k$-th roots of unity for some $k$, where $p$ is a fixed prime.

Answer 2

You are asking for groups in which every non-trivial subgroup is essential.

We can classify all abelian groups with this property, as follows.

Embed $G$ in its injective envelope $D$, which is a divisible group that is an essential extension of $G$. If $H,K$ are non-trivial subgroups of $D$, then $H\cap G$ and $K\cap G$ are non-trivial subgroups of $G$ (here we use that it is an essential extension). But then their intersection $(H\cap G)\cap (K\cap G)$ is nontrivial, so $H\cap K$ is non-trivial.

So $D$ is also an abelian group with NIP. Since $D$ is divisible, by the structure theorem it is a direct product of copies of $\mathbb{Q}$ and Prüfer groups . Since a group with NIP cannot be a direct product, $D$ is either equal to $\mathbb{Q}$, or a Prüfer group.

So if $G$ is torsion-free, it is a subgroup of $\mathbb{Q}$. If it has torsion, then it is a subgroup of a Prüfer group, which means that it is either cyclic of prime power order, or a Prüfer group.

The non-abelian case appears to be substantially more difficult. As Jyrki Lahtonen points out, the quaternion group $Q_8$ is a finite example; it would be nice to know if there are infinite examples.

One observation: a group $G$ with NIP that has a torsion element must have a unique minimal subgroup of order $p$. I believe that it is known that such a $G$ must be abelian if $p>2$, but I don't know if there is a simple proof of this, or a characterization of the non-abelian $2$-groups with unique minimal subgroup. (studiosus and Jyrki mention some infinite classes of examples)