# p-Group satisfying the minimal condition on abelian subgroups

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.

• Restatement: find a $p$-group with a properly decreasing sequence of subgroups, but not containing any copy of $(\mathbf{Z}/p\mathbf{Z})^{(\infty)}$.
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:
• 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.