Cyclic Groups

A cyclic group $G$ is a group that can be generated by a single element $a$, the so-called group generator, which we denote by $\langle a\rangle$. To be precise, if $a$ is a generator of the group $G$, then $G$ consists of all the powers of $a$, i.e.,

$$\langle a\rangle=\left\{\ldots, a^{-2}, a^{-1}, a^{0}=1=e_{G}, a, a^{2}, a^{3}, \ldots\right\}$$

thus every element in $G$ has the form $a^{i}$ for some integer $i \in \mathbb{Z}$, where

$$a^{n+1}=a^{n} * a$$

and we say that $G$ is a cyclic group generated by $a$. Hence, a group $G$ is said to be cyclic if there exists an element in $G$ whose powers generate the whole group.

Cyclic group

Definition:

A group $G$ is said to be cyclic if there exists an element $a \in G$ such that

$$G=\langle a\rangle=\left\{a^{n} \mid n \in \mathbb{Z}\right\},$$

where $a$ is called the generator of $G$.

Note: A cyclic group may have many different generators.

Corollary 1

Every cyclic group is abelian.

Proof:

Let x, y \in G=\langle a\ranglegle . We denote $x=a^{i}$ and $y=a^{j} .$ Then, clearly

$$x \cdot y=a^{i} a^{j}=a^{i+j}=a^{j+i}=a^{j} a^{i}=y \cdot x.$$

Example:

The following examples show cyclic groups:

1. The set $\mathbb{Z}$ of integers under addition is an infinite cyclic group, where $\mathbb{Z}=\langle 1\rangle=\langle-1\rangle$.

2. The set $\mathbb{Z}_{n}=\{0,1,2, \ldots, n-1\}$ under addition modulo $n$ is a finite cyclic group, where $\mathbb{Z}_{n}=\langle 1\rangle=\langle n-1\rangle$ because $n-1 \equiv-1 \space\space\space mod \space n$. Other generators are possible depending on $n$. For example, $\mathbb{Z}_{10}=\langle 1\rangle=$ $\langle 9\rangle=\langle 3\rangle=\langle 7\rangle$.

$Part(2)$ of the example shows an important fact, explained in the following sections.

Lemma 1

If $a$ is a group generator, then so it’s its inverse $a^{-1}$, meaning that if $G=\langle a\rangle$, then also $G=\left\langle a^{-1}\right\rangle$.

Proof:

Suppose that $G=\langle a\rangle.$ Let $g$ be any group element, i.e., $g \in G .$ Since $G=\langle a\rangle$, there exists an $n \in \mathbb{Z}$ such that $g=a^{n}$.

Then, $g^{-1}=\left(a^{n}\right)^{-1}=a^{-n}=\left(a^{-1}\right)^{n} \in\left\langle a^{-1}\right\rangle$.

Since $\left\langle a^{-1}\right\rangle$ is closed under inverses and $g^{-1} \in\left\langle a^{-1}\right\rangle$, it follows that $g=\left(g^{-1}\right)^{-1} \in\left\langle a^{-1}\right\rangle$, hence $G \subseteq\left\langle a^{-1}\right\rangle$, and so $G=\left\langle a^{-1}\right\rangle$.

Lemma 2

Let $G=\langle a\rangle$ be a cyclic group. Then one of the following holds:

1. $\langle a\rangle \cong(\mathbb{Z},+)$ if $G$ is an infinite cyclic group.

2. $\langle a\rangle \cong\left(\mathbb{Z}_{n},+\right)$, where $n$ is the smallest positive integer such that $a^{n}=e_{G}$.

The following example is of great importance in order to understand the properties of cyclic groups.

Example:

Consider the cyclic group $g$ of order $12$ with generator $a$. Then, the group consists of the following set:

$$G=\left\{1, a^{1}, a^{2}, a^{3}, a^{4}, a^{5}, a^{6}, a^{7}, a^{8}, a^{9}, a^{10}, a^{11}\right\}.$$

We observe that

$$\left\langle a^{5}\right\rangle=\left\{a^{5}, a^{10}, a^{3}, a^{8}, a, a^{6}, a^{11}, a^{4}, a^{9}, a^{2}, a^{7}, 1\right\}=G$$

,hence $a^{5}$ is also a group generator next to $a$. By using the same argument, $a^{7}$ and $a^{11}$ also generate $G$, as discussed earlier. Any other element of the group is not a generator of $G$, as

$$\begin{aligned} \langle 1\rangle &=\{1\}, \\ \left\langle a^{6}\right\rangle &=\left\{1, a^{6}\right\}, \\ \left\langle a^{4}\right\rangle=\left\langle a^{8}\right\rangle &=\left\{1, a^{4}, a^{8}\right\}, \\ \left\langle a^{3}\right\rangle=\left\langle a^{9}\right\rangle &=\left\{1, a^{3}, a^{6}, a^{9}\right\}, \\ \left\langle a^{2}\right\rangle=\left\langle a^{10}\right\rangle &=\left\{1, a^{2}, a^{4}, a^{6}, a^{8}, a^{10}\right\} . \end{aligned}$$