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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

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  • $\begingroup$ Restatement: find a $p$-group with a properly decreasing sequence of subgroups, but not containing any copy of $(\mathbf{Z}/p\mathbf{Z})^{(\infty)}$. $\endgroup$
    – YCor
    Oct 31, 2016 at 17:32

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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.
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