Question

Give any two examples of a non-cyclic group, all of whose proper subgroups are

cyclic.

Answer 1

$\text{An example of a non-cyclic subgroup is U(8), since none of its elements i.e}\\ \{1,3,5,7\} \text{ is a generator of U(8), but the only proper subgroups of U(8), $\{1\}, \{1,3\}$}\\ \text{$\{1,5\}, \{1,7\}$ are all cyclic subgroups}\\ \text{Another example is Q8 = $\{1,i,j,k,-1,-i,-j,-k\}$ which is non cyclic but its proper }\\ \text{subgroups are all cyclic}$