TITLE: Concerning Groups having the property that intersection of any two non-trivial subgroups is non-trivial
QUESTION [11 upvotes]: 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

REPLY [2 votes]: As examples of finite groups with this property let me add the (generalized) quaternion groups. The quaternion group $Q_8=\{\pm1,\pm i,\pm j,\pm k\}$ occurs often enough in introductory courses. It has the given property because $-1$ is the only element of order two, and thus is contained in every non-trivial subgroup.
The generalied version $Q_{2^n}, n>3,$ is defined by 
$$
Q_{2^n}=\langle a,b\mid a^{2^{n-2}}=b^2, b^4=1, bab^{-1}=a^{2^{n-2}+1}\rangle.
$$
It is easier for me to think of this as the group generated by the complex matrices
$$
A=\left(\begin{array}{cc}\zeta&0\\0&\zeta^{-1}\end{array}\right),
\qquad
B=\left(\begin{array}{cc}0&1\\-1&0\end{array}\right),
$$
where $\zeta$ is a primitive root of unity of order $2^{n-1}$.
Again, the element $-I_2=B^2=A^{2^{n-2}}$ is the only element of order two, and is thus contained in all the non-trivial subgroups.