Prove that all subgroup of a cyclic group is cyclic.
Let be a cyclic group, a subgroup of , and such that .
Let be the only morphism of groups such that .
Then is a subgroup of , so it's a cyclic group.
So (because of surjectivity) is a cyclic group.
Why, if is a cyclic group, another group, and a morphism of groups between and , then is cyclic?
Let be a generator of , and an element of . There is a such that . Thus