A cyclic group is a group that is generated by a single element, called its generator. Each element of the group can be written as repeated group operations on the generator with itself.

Formally, for any element $g$ in any group $G$, the cyclic subgroup generated by $g$ is written

Properties

A group element of finite order generates a finite cyclic subgroup of that order.
A group element of infinite order generates an infinite cyclic subgroup.
Cyclic subgroups are equivalent if one power is expressible as the gcd of the other with the order of the base.
Fundamental Theory of Cyclic Groups:

• Every subgroup of a cyclic group is itself cyclic.
• The order of any cyclic subgroup divides the order of any cyclic group which contains it.
• There is exactly one subgroup of the order of each divisor of the order of its container cyclic group.