A cyclic group is a group that can be generated by just one element $a$.

If a group $G$ contains an element $a$ such that every element $x \in G$ is of the form $a ^ {m}$, where $m \in Z$, then $G$ is said to be a cyclic group and $G$ is generated by $a$. In this case we write:

If $a$ is a generator of $G$, then $a ^ {-1}$ is also a generator of $G$.

But let's make an example: the multiplicative group $G = \{ 1, -1, i, -i \}$ is a cyclic group of order $4$ with generator $i$, i.e.

Some of the most fundamental properties of cyclic groups are listed below:

1. Every cyclic group is abelian, but abelian groups don't need to be cyclic
2. Every subgroup of a cyclic group is cyclic but the group $G$ is not necessarily cyclic if all its subgroups are cyclic
3. The number of generators of a finite cyclic group of order $n$ is $\phi (n)$, where $\phi (n)$ is the number of integers less than $n$ and coprime to $n$
4. If a cyclic group $G$ is generated by an element $a$ of order $n$, then $a ^ {m}$ is a generator of $G$ if and only if $gcd(m, n) = 1$
5. Every group of order $p$, with $p$ prime, is cyclic
6. A cyclic group of finite order $n$ is isomorphic to the additive group of residue classes modulo $n$