## Section4.1Cyclic groups

Often a subgroup will depend entirely on a single element of the group; that is, knowing that particular element will allow us to compute any other element in the subgroup.

##### Remark4.1.4

If we are using the “+” notation, as in the case of the integers under addition, we write $\langle a \rangle = \{ na : n \in {\mathbb Z} \}\text{.}$

For $a \in G\text{,}$ we call $\langle a \rangle$ the cyclic subgroup generated by $a\text{.}$ If $G$ contains some element $a$ such that $G = \langle a \rangle \text{,}$ then $G$ is a cyclic group. In this case $a$ is a generator of $G\text{.}$ If $a$ is an element of a group $G\text{,}$ we define the order of $a$ to be the smallest positive integer $n$ such that $a^n= e\text{,}$ and we write $|a| = n\text{.}$ If there is no such integer $n\text{,}$ we say that the order of $a$ is infinite and write $|a| = \infty$ to denote the order of $a\text{.}$

The groups ${\mathbb Z}$ and ${\mathbb Z}_n$ are cyclic groups. The elements 1 and $-1$ are generators for ${\mathbb Z}\text{.}$ We can certainly generate ${\mathbb Z}_n$ with 1 although there may be other generators of ${\mathbb Z}_n\text{,}$ as in the case of ${\mathbb Z}_6\text{.}$