Cyclic group

Equivalence of definitions
The second and third definition are equivalent because the subgroup generated by an element is precisely the set of its powers. The first definition is equivalent to the other two, because:


 * The image of $$1 \in \mathbb{Z}$$ under a surjective homomorphism from $$\mathbb{Z}$$ to $$G$$ must generate $$G$$
 * Conversely, if an element $$g$$ generates $$G$$, we get a surjective homomorphism $$\mathbb{Z} \to G$$ by $$n \mapsto g^n$$

Arithmetic functions
See finite cyclic group and group of integers.

Facts

 * There is exactly one cyclic group (upto isomorphism of groups) of every positive integer order $$n$$: namely, the group of integers modulo $$n$$. There is a unique infinite cyclic group, namely $$\mathbb{Z}$$
 * For any group and any element in it, we can consider the subgroup generated by that element. That subgroup is, by definition, a cyclic group. Thus, every group is a union of cyclic subgroups.