Definition:Cyclic Group

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Definition

Definition 1

The group $G$ is cyclic if and only if every element of $G$ can be expressed as the power of one element of $G$:

$\exists g \in G: \forall h \in G: h = g^n$

for some $n \in \Z$.


Definition 2

The group $G$ is cyclic if and only if it is generated by one element $g \in G$:

$G = \gen g$


Generator

Let $a \in G$ be an element of $g$ such that $\gen a = G$.

Then $a$ is a generator of $G$.


Notation

A cyclic group with $n$ elements is often denoted $C_n$.

Some sources use the notation $\sqbrk g$ or $\gen g$ to denote the cyclic group generated by $g$.


From Integers Modulo m under Addition form Cyclic Group, $\struct {\Z_m, +_m}$ is a cyclic group.

Thus $\struct {\Z_m, +_m}$ often taken as the archetypal example of a cyclic group, and the notation $\Z_m$ is used.

This is justified as, from Cyclic Groups of Same Order are Isomorphic, $\Z_m$ is isomorphic to $C_m$.

In certain contexts $\Z_m$ is particularly useful, as it allows results about cyclic groups to be demonstrated using number theoretical techniques.


Examples

Subgroup of $\struct {\R_{\ne 0}, \times}$ Generated by $2$

Consider the multiplicative group of real numbers $\struct {\R_{\ne 0}, \times}$.

Consider the subgroup $\gen 2$ of $\struct {\R_{\ne 0}, \times}$ generated by $2$.

Then $\gen 2$ is an infinite cyclic group.


Subgroup of $\struct {\C_{\ne 0}, \times}$ Generated by $i$

Consider the multiplicative group of real numbers $\struct {\C_{\ne 0}, \times}$.

Consider the subgroup $\gen i$ of $\struct {\C_{\ne 0}, \times}$ generated by $i$.

Then $\gen i$ is an (finite) cyclic group of order $4$.


Group Presentation

The presentation of a finite cyclic group of order $n$ is:

$C_n = \gen {a: a^n = e}$


Also see

  • Results about cyclic groups can be found here.


Sources