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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?

2 Answers 2

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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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    @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.2012-11-04
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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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    Yes, edited. My computer started to reboot just before I wanted to edit this. Thanks. Back online again.2012-11-04