Cyclic Groups
Basic definitions and examples

Let $G$ be a group and $g\in G$. Then \(< g > = \{g^{k} : k\in\mathbb{Z}\}\) is a subgroup of $G$ called the cyclic subgroup of $G$ generated by $g$.

A group $G$ is cyclic if there is $g\in G$ such that $G= < g >$. In this case, we say that $g$ generates $G$.

Examples of cyclic groups: $\mathbb{Z}$, $\mathbb{Z}_{n}$, $U(7)$, rotations of an equilateral triangle.

Examples of noncyclic groups: Symmetries of an equilateral triangle, quaternion group.
Properties of cyclic subgroups and groups

The subset generated by an element of a group is a subgroup.

Cyclic groups are abelian.

Every subgroup of a cyclic group is cyclic.
Orders of elements
Note: Please review Euclid’s Algorithm. See video and notes for the theory and this video for a numerical example.
You may also find it helpful to look at how one uses Euclid’s algorithm to solve congruence equations.
See video and notes for discussion of this material.

The order of an element $g\in G$ is the number of elements in $\langle g \rangle$.

Suppose that $G$ is a cyclic group of order $n$, and that $a$ is a generator of $G$. Then $a^{k}=e$ if and only if $n$ divides $k$.

Let $G$ be a cyclic group of order $n$ and suppose that $a$ is a generator of $G$. If $b=a^{k}$, then the order of $b$ is $n/d$ where $d=\mathrm{gcd}(k,n)$.

A congruence class $[r]$ generates $\mathbb{Z}_{n}$ if and only if $\mathrm{gcd}(r,n)=1$. More generally, if $G$ is a cyclic group of order $n$ generated by $g$, then $g^r$ is a generator of $G$ if and only if $\mathrm{gcd}(r,n)=1$.
Further examples and results.
 For any $n>0$, the complex solutions to the polynomial $z^{n}=1$ form a cyclic group of order $n$ called the group of $n^{th}$ roots of unity. A generator of the group of $n^{th}$ roots of unity is called a primitive $n^{th}$ root of unity.