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?
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?
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.
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.