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