Cyclic Group

NAVIGATION

Home

Research

Bookshelf

Garden

FIND ME ON

GitHub

Home

Research

Bookshelf

Garden

Theorem (Cyclic group)

A cyclic group is a group which is equal to one of its cylic subgroups i.e. $\exists g\in G$ s.t. $G=\langle g\rangle$ where $g$ is called the generator of $G$ .

Definition (Cyclic subgroup)

For a group $G$ , any element $g\in G$ one can form a subgroup of all its integer powers $\langle g\rangle=\{ g^{k}: k\in\mathbb{Z} \}$ called the cyclic subgroup generated by g.

Theorem (Finite Subgroups are Cyclic)

Let $\mathbb{F}$ be any field. Any finite subgroup $G$ of $\mathbb{F}^{*}$ is cyclic.

Cor

The only finite subgroups of $\mathbb{C}^{*}$ are the groups $\mu_{n}$ consisting of the $n$ -th roots of unity with $n\ge 1$ .

Cor

If $\mathbb{F}$ is a finite field, then $\mathbb{F}^{*}$ is cyclic.

Linked from

Cyclic Group

Theorem 3.1