Definition – Cyclic Group – Math-reference.net

Contents

1. Introduction
2. Definitions

Definition

Given a group, $G$ , with some group operation $\times$ , we use the notation

$g^i = \underbrace{g\times g \times ... \times g}_{\text{i times}}$

Ie, as shorthand for the number of times we apply the operation to the element.

Let $G$ be a finite group with $|G| =n$ . If there exists an element $g\in G$ such that every element $x\in G$ can be expressed in the form $g^{i~(mod~n)}= x$ for some $i$ , and $g^{i'} = g^{j'} \implies i'=j'$ , we refer to $G$ as a cyclic group. Ie, we must be able to write the elements of the group as

$G = \{e_G, g^1, ..., g^{n-1}\},\quad e_G=g^0$

If we can write a group in this way, we say that the group $G$ is generated by $g\in G$ , and we use the notation $G = \langle g \rangle$ .

If $G$ is an infinite group, then we must be able to write the elements of the group in the form,

$G = \{..., g^{-2}, g^{-1}, e_G, g^1, g^2, ...\},\quad e_G=g^0$

Again, in this case we still write $G = \langle g \rangle$ .