If all Subgroups are Cyclic, is group Cylic? I am having difficulty seeing if it is the case that if all subgroups of a group are cyclic, that the group itself is cyclic.
 A: No this does not have to be the case - take $V_4$ or $S_3$, the Klein 4-group or the symmetric group on 3 symbols respectively. So the whole group does not have to be even abelian.
A: In addition to the finite groups mentioned above, there are also infinite examples. The construction is nontrivial, but a Tarski monster is an infinite group $G$ with every proper subgroup $H\subset G$ cyclic of order $p$ for some fixed prime $p$ (independent of $H$). They exist for sufficiently large $p$.
A: The group itself is a subgroup, so that it is cyclic as well, because all subgroups are cyclic. Did you maybe mean that all strict subgroups are cyclic ? In this case, it is wrong to conclude that the group itself is cyclic : for instance $G = \left\{ \frac{x}{2^d}\;|\;x\in\mathbf{Z}, d\in\mathbf{n}\right\}$ is not cyclic, and has only cyclic strict subgroups. (I am giving this example, as all given examples were finite groups.)
A: If you mean proper subgroups, then No! consider $G = \mathbf{Z}_2 \times \mathbf{Z}_2$.
