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)
$e$	$e$	$x$	$x^{2}$
$x$	$x$	$x^{2}$	$e$
$x^{2}$	$x^{2}$	$e$	$x$

Arithmetic function

Function	Value	Explanation
Order	3
Exponent	3
Derived length	1	The group is an abelian group.
Nilpotency class	1	The group is an abelian group.
Frattini length	1
Fitting length	1		number of subgroups	2
number of conjugacy classes	3
number of conjugacy classes of subgroups	2

Group properties

Property	Satisfied	Explanation	Comment
Group of prime order	Yes	By definition
Cyclic group	Yes	By definition	Smallest odd-order cyclic group
Elementary abelian group	Yes	By definition
Abelian group	Yes	Cyclic implies abelian
Nilpotent group	Yes	Abelian implies nilpotent
Metacyclic group	Yes	Cyclic implies metacyclic
Supersolvable group	Yes	Cyclic implies supersolvable
Solvable group	Yes	Abelian implies solvable
T-group	Yes
Simple group	Yes	Simple abelian if and only if prime order

Endomorphisms

Any endomorphism of a cyclic group is determined by where it sends the generator. The cyclic group of order three has three endomorphisms:

The identity map is an endomorphism. This map sends every element to itself.
The square map is an endomorphism. This map sends $x$ to $x^{2}$ , $x^{2}$ to $x$ , and $e$ to itself.
The trivial map is an endomorphism. This map sends every element to $e$ .

Automorphisms

Of the three endomorphisms, two are automorphisms: the identity map and the square map. These form a cyclic group of order two: the square map, applied twice, gives the identity map.

Subgroups

The group has no proper nontrivial subgroups: the only subgroups are the whole group and the trivial subgroup.

More generally, any nontrivial group with no proper nontrivial subgroup must be cyclic of prime order. Conversely, any cyclic group of prime order has no proper nontrivial subgroup. Further information: No proper nontrivial subgroup implies cyclic of prime order

Quotients

The cyclic group of order three has only two quotients: the whole group and the trivial quotient. This follows from the fact that this group is simple -- it has no proper nontrivial normal subgroup.

Other constructions

Template:Holomorph

The holomorph of this group, i.e., the semidirect product of this group with its automorphism group, is isomorphic to the symmetric group on three letters. The cyclic group sits inside this as the alternating group and the automorphism group sits inside as a subgroup of order two.

In larger groups

Occurrence as a subgroup

The cyclic group of order 3 occurs as a subgroup in many groups. In general, a group contains a cyclic subgroup of order three if and only if its order is a multiple of three (this follows from Cauchy's theorem, a corollary of Sylow's theorem).

Occurrence as a normal subgroup

The cyclic group of order three occurs as a normal subgroup in some groups.

For instance, if a field contains non-identity cuberoots of unity, then the multiplicative group of the field contains a cyclic subgroup of order three. As a corollary, the general linear group contains a central subgroup of order three.

Normal subgroups of order three need not be central; for instance, in the symmetric group on three letters, the alternating group is a normal subgroup of order three, but is not central. However, for an odd-order group, any normal subgroup of order three is central. This follows from a more general fact: a normal subgroup whose order is the least prime divisor of the order of the group is central.

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)

It can also be described as:

CyclicGroup(3)

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