📂Abstract Algebra

Cyclic groups in Abstract Algebra

Definition ¹

A group $G$, having an element $a$ and for any $x \in G$ there exists an integer $n \in \mathbb{Z}$ satisfying $x = a^{n}$, is called a Cyclic group, and $a$ is called a Generator.

Explanation

Simply put, if all elements of a group can be expressed as the power of a generator, then it is a cyclic group. The term ‘cyclic’ is quite apt since it involves continuously expressing all elements as powers of the generator.

Not immediately apparent from the definition is that all cyclic groups are abelian groups, and the generator is not necessarily unique. Theorem [1] is an example of this.

Furthermore, according to the definition, cyclic groups do not have to be finite groups. What is important to note is that $n$ exists but is an integer, not a natural number, which means that adding the inverse of the generator is also fine. Theorem [2] is an example of this.

Theorems

• [1]: The generators of $\mathbb{Z}_{4} = \left\{ 0,1,2,3 \right\}$ are not unique.
• [2]: $\mathbb{Z}$ is a cyclic group.

Proof

[1]

Even though $1$ alone can represent all the elements, since $3 \equiv -1 \pmod{4}$, $3$ can also represent all elements.

Therefore, $\left< 1 \right> = \left< 3 \right> = \mathbb{Z}_{4}$, and it can be understood that a generator does not have to be unique.

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[2]

Since all elements of $\left< \mathbb{Z} , + \right>$ can be represented as $1 \cdot n = (-1) \cdot (-n) = n$, thus $\mathbb{Z} = \left< 1 \right> = \left< -1 \right>$

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