Cyclic group

by Patrick Stevens Jun 13 2016 updated Jul 10 2016

Cyclic groups form one of the most simple classes of groups.

[summary: The cyclic groups are the simplest kind of group; they are the groups which can be made by simply "repeating a single element many times". For example, the rotations of a polygon.]

[summary(technical): A group $~$G$~$ is cyclic if it has a single [generator_mathematics generator]: there is one element $~$g$~$ such that every element of the group is a [ power] of $~$g$~$.]


A cyclic group is a group $~$(G, +)$~$ (hereafter abbreviated as simply $~$G$~$) with a single generator, in the sense that there is some $~$g \in G$~$ such that for every $~$h \in G$~$, there is $~$n \in \mathbb{Z}$~$ such that $~$h = g^n$~$, where we have written $~$g^n$~$ for $~$g + g + \dots + g$~$ (with $~$n$~$ terms in the summand). That is, "there is some element such that the group has nothing in it except powers of that element".

We may write $~$G = \langle g \rangle$~$ if $~$g$~$ is a generator of $~$G$~$.



Cyclic groups are abelian

Suppose $~$a, b \in G$~$, and let $~$g$~$ be a generator of $~$G$~$. Suppose $~$a = g^i, b = g^j$~$. Then $~$ab = g^i g^j = g^{i+j} = g^{j+i} = g^j g^i = ba$~$.

Cyclic groups are countable

The elements of a cyclic group are nothing more nor less than $~$\{ g^0, g^1, g^{-1}, g^2, g^{-2}, \dots \}$~$, which is an enumeration of the group (possibly with repeats).