Prove or disprove If a group G is such that every proper sub

Prove or disprove: If a group G is such that every proper subgroup is cyclic, then G is cyclic.

Solution

All of its proper subgroups are cyclic, but V₄ is not itself cyclic. Another example would be the dihedral group D₃. Every proper subgroup has order 1, 2, or 3, and is thus cyclic. However, D₃ is not cyclic. (It is not even abelian.)

Of course, the counterexample can be shown: the Klein group Z/2ZZ/2Z.