Are there examples of $p$-groups satisfying the minimal condition on abelian subgroups but do not satisfying the minimal condition on subgroups?


Obviously such a group cannot be locally finite.


I've already asked it on stackexchange but no answer in a few days.
<https://math.stackexchange.com/questions/1981177/p-group-satisfying-the-minimal-condition-on-abelian-subgroups>



p-Group satisfying the minimal condition on abelian subgroups
The free Burnside group $B(2,n)$, of rank $2$ and exponent $n$, where $n$ is a sufficiently large power of an odd prime, satisfies both conditions: 


* all of its abelian subgroups are cyclic (S. I. Adian, "The Burnside Problem and Identities in Groups," Nauka, Moscow, 1975;
English transl.: Springer-Verlag, New York, 1979), which implies the minimality condition on abelian subgroups;
* it contains a copy of the free Burnside group $B(\infty,n)$, of infinite countable rank (V. L. Shirvanyan, "Embedding the group $B(\infty,n)$ in the group $B(2,n)$", Mathematics of the USSR-Izvestiya, 1976, 10:1, 181–199), hence it does not have the minimality condition on all subgroups.


