Definition:Cyclic Group
This page is about Cyclic Group. For other uses, see Cyclic.
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 complex 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
- Group Generated by Singleton
- List of Elements in Finite Cyclic Group
- Order of Cyclic Group equals Order of Generator
- Results about cyclic groups can be found here.