Cyclic group:Z3

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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
The multiplicative group of the field of four elements. In particular, it is the general linear group $GL(1,4)$ .

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 functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 3#Arithmetic functions

Function	Value	Similar groups	Explanation for function value
underlying prime of p-group	3
order (number of elements, equivalently, cardinality or size of underlying set)	3	groups with same order
prime-base logarithm of order	1	groups with same prime-base logarithm of order
max-length of a group	1		max-length of a group equals prime-base logarithm of order for group of prime power order
chief length	1		chief length equals prime-base logarithm of order for group of prime power order
composition length	1		composition length equals prime-base logarithm of order for group of prime power order
exponent of a group	3	groups with same order and exponent of a group \| groups with same prime-base logarithm of order and exponent of a group \| groups with same exponent of a group
prime-base logarithm of exponent	1	groups with same order and prime-base logarithm of exponent \| groups with same prime-base logarithm of order and prime-base logarithm of exponent \| groups with same prime-base logarithm of exponent
Frattini length	1	groups with same order and Frattini length \| groups with same prime-base logarithm of order and Frattini length \| groups with same Frattini length	Frattini length equals prime-base logarithm of exponent for abelian group of prime power order
minimum size of generating set	1	groups with same order and minimum size of generating set \| groups with same prime-base logarithm of order and minimum size of generating set \| groups with same minimum size of generating set
subgroup rank of a group	1	groups with same order and subgroup rank of a group \| groups with same prime-base logarithm of order and subgroup rank of a group \| groups with same subgroup rank of a group	same as minimum size of generating set since it is an abelian group of prime power order
rank of a p-group	1	groups with same order and rank of a p-group \| groups with same prime-base logarithm of order and rank of a p-group \| groups with same rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
normal rank of a p-group	1	groups with same order and normal rank of a p-group \| groups with same prime-base logarithm of order and normal rank of a p-group \| groups with same normal rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
characteristic rank of a p-group	1	groups with same order and characteristic rank of a p-group \| groups with same prime-base logarithm of order and characteristic rank of a p-group \| groups with same characteristic rank of a p-group	same as minimum size of generating set since it is an abelian group of prime power order
nilpotency class	1		The group is a nontrivial abelian group
derived length	1		The group is a nontrivial abelian group
Fitting length	1		The group is a nontrivial abelian group -

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.

GAP implementation

Group ID

This finite group has order 3 and has ID 1 among the groups of order 3 in GAP's SmallGroup library. For context, there are groups of order 3. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(3,1)

For instance, we can use the following assignment in GAP to create the group and name it $G$ :

gap> G := SmallGroup(3,1);

Conversely, to check whether a given group $G$ is in fact the group we want, we can use GAP's IdGroup function:

IdGroup(G) = [3,1]

or just do:

IdGroup(G)

to have GAP output the group ID, that we can then compare to what we want.

Other descriptions

The cyclic group of order three can be defined using the CyclicGroup function:

CyclicGroup(3)

It can also be defined as the alternating group of degree three, using the AlternatingGroup function:

AlternatingGroup(3)

Internal links