Let $G$ be a finite group with the following property:


For every nontrivial proper subgroups $H$ and $K$ for which $H\cap K=1$, if the number of their minimal subgroups are $m$ and $n$ respectively, then the number of minimal subgroups of the subgroup generated by $H$ and $K$,$\langle H,K\rangle$, is $m+n$. 


What are the possible structures for $G$? 
I found only cyclic groups and direct product product of a generalized quaternion group and a cyclic group of some odd order. Is there any other case?



Minimal Subgroups
We show that $G$ satisfies the given properties iff $G$ has a unique subgroup of order $p$ for every prime divider $p$ of $|G|$ and that every two Sylow subgroups of distinct orders permute.


Necessity: If $A$ and $B$ are distinct cyclic subgroups of prime order $p$, then $[A,B]\neq1$ otherwise $\langle A,B\rangle\cong C\_p\times C\_p$ has $p+1>2$ minimal subgroups, which contradicts the assumption. But then $\langle A,B\rangle$ has at least $p+1$ minimal subgroups including the conjugates of $A$ by elements of $B$ and $B$ itself, which is another contradiction. This shows that $G$ has a unique subgroup of order $p$ for every prime $p$ dividing $|G|$. Now if $P$ and $Q$ are a Sylow $p$-subgroup and a Sylow $q$-subgroup of $G$, respectively, with $p\neq q$, then $P\cap Q=1$ and since by assumption $\langle P,Q\rangle$ is a $\{p,q\}$-group, we must have $\langle P,Q\rangle=PQ=QP$, as required.


Sufficiency: Let $H$ and $K$ be two subgroups satisfying $H\cap K=1$. Since $G$ has a unique subgroup of any prime order dividing $|G|$, we must have $\gcd(|H|,|K|)=1$. Then $H\subseteq\prod\_{i\in\pi(H)}P\_i$ and $K\subseteq\prod\_{i\in\pi(K)}P\_i$ for suitable Sylow subgroups $P\_i$. Since $P\_i$ and $P\_j$ permute pairwise, we have $\langle H,K\rangle\subseteq\prod\_{i\in\pi(H)\cup\pi(K)}P\_i$, from which the result follows.


Note that by a theorem of Suzuki all these groups are solvable (See [MR0074411](http://www.ams.org/mathscinet-getitem?mr=0074411)).


