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Prove or disprove?

I'm leaning towards it being true but not sure. Any hint would be greatly appreciated.

In case of it being false, i.e a non-cyclic p-group can have all cyclic proper subgroups, is there any way to count them?

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If you take the smallest non-cyclic $p$-group, then its proper subgroups are smaller $p$-groups and thus have to be cyclic. So, it can happen.

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And if it is not the smallest non-cyclic p-group, i.e if I have already proven there is one of smaller order? – ikk Nov 4 '12 at 10:09
@ikk Do you understand that a single counterexample disproves the statement in you title? As far as your new question goes, I suggest that you identify the smallest non-cyclic $p$-group and check whether it is a subgroup of all other non-cyclic $p$-groups. – Phira Nov 4 '12 at 10:17

Take $V_4$, the Klein group of order 4 or the quaternion group $Q$ of order 8, or the dihedral group $D_4$ of order 8. Those are the smallest examples of non-cyclic groups with only proper cyclic subgroups.

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Erm - isn't the Klein 4-group the smallest non-cyclic group with all of its proper subgroups cyclic? – Old John Nov 4 '12 at 15:25
Yes, edited. My computer started to reboot just before I wanted to edit this. Thanks. Back online again. – Nicky Hekster Nov 4 '12 at 15:37

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