Cyclic group:Z3

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This particular group is a finite group of order: 3

This particular group is the smallest (in terms of order): non-ambivalent group

This particular group is the smallest (in terms of order): odd-order group

Definition

Verbal definition

The cyclic group of order 3 is defined as the unique group of order 3. Equivalently it can be described as a group with three elements $e=x^0,x,x^2$ where $x^l{x}^m=x^{l+m}$ with the exponent reduced mod 3. It can also be viewed as:

• The quotient group of the group of integers by the subgroup of multiples of 3
• The multiplicative group comprising the three cuberoots of unity (as a subgroup of the multiplicative group of nonzero complex numbers)
• The alternating group on three elements
• The group of orientation-preserving symmetries (rotational symmetries) of the equilateral triangle

Multiplication table

Element	$e$ (identity element)	$x$ (generator)	$x^2$ (generator)
$e$	$e$	$x$	$x^2$
$x$	$x$	$x^2$	$e$
$x^2$	$x^2$	$e$	$x$

Properties

Cyclicity

This particular group is cyclic

Abelianness

This particular group is Abelian

Nilpotence

This particular group is nilpotent

Solvability

This particular group is solvable

Simplicity

This particular group is simple: it has no proper nontrivial normal subgroup

Implementation in GAP

Group ID

Since the cyclic group of order 3 is the only group of order 3, its ID is 1. So it can be described as:

SmallGroup(3,1)

Internal links

• Linear representation theory of cyclic group:Z3