Trivial group: Difference between revisions

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{{basicdef}}
{{particular group}}
{{particular group}}
[[Category:Symmetric groups]]


<section begin=beginner/>
==Definition==
==Definition==


The '''trivial group''' is the group with only one element, which is its identity element.
===Verbal definition===
 
The '''trivial group''' is the group with only one element, which is its identity element. The trivial group is usually denoted as <math>1</math>, <math>\{ 1 \}</math>, or <math>\{ e \}</math>.
<section end=beginner/>
<section begin=revisit/>


===Alternative definitions===
===Alternative definitions===
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* The {{cyclic group}} on one element
* The {{cyclic group}} on one element
* The {{symmetric group}} on one element
* The {{symmetric group}} on one element
* The {{alternating group}} on one or two elements
* The {{projective general linear group}} of order 1 over any field
* The {{projective general linear group}} of order 1 over any field
* The {{special linear group}} of order 1 over any field
* The {{special linear group}} of order 1 over any field
* The {{general linear group}} <math>GL(1,2)</math>
* The {{orthogonal group}} of order 1 over a field of characteristic two
===Multiplication table===
{| class="wikitable" border="1"
!Element !! <math>e</math> (identity element)
|-
| <math>e</math> || <math>e</math>
|}


==Importance==
==Importance==
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The trivial group is important in the following ways:
The trivial group is important in the following ways:


* For any group, there is a unique homomorphism from the trivial group to that group, namely the homomorphism sending it to the identity element. Thus, the trivial group occurs in a unique way as a subgroup for any given group, namely the one-element subgroup comprising the identity element. This is termed the '''trivial subgroup'''.
* For any group, there is a unique homomorphism from the trivial group to that group, namely the homomorphism sending it to the identity element. Thus, the trivial group occurs in a unique way as a [[subgroup]] for any given group, namely the one-element subgroup comprising the identity element. This is termed the '''trivial subgroup'''.


* For any group, there is a unique homomorphism to the trivial group from that group, namely the homomorphism sending everything to the identity element. Thus, the trivial group occurs in a unique way as a [[quotient group]] of any given group, namely its quotient by itself. This is termed the '''trivial quotient'''.
* For any group, there is a unique homomorphism to the trivial group from that group, namely the homomorphism sending everything to the identity element. Thus, the trivial group occurs in a unique way as a [[quotient group]] of any given group, namely its quotient by itself. This is termed the '''trivial quotient'''.
<section end=revisit/>
==Arithmetic functions==
{| class="wikitable" border="1"
! Function !! Value !! Explanation
|-
| [[order of a group|order]] || [[arithmetic function value::order of a group;1|1]] || only the identity element.
|-
| [[exponent of a group|exponent]] || [[arithmetic function value::exponent of a group;1|1]] ||
|-
| [[nilpotency class]] || [[arithmetic function value::nilpotency class;0|0]] ||
|-
| [[derived length]] || [[arithmetic function value::derived length;0|0]] ||
|-
| [[Frattini length]] || [[arithmetic function value::Frattini length;0|0]] ||
|-
| [[Fitting length]] || [[aritmhetic function value::Fitting length;0|0]] ||
|-
| [[minimum size of generating set]] || [[arithmetic function value::minimum size of generating set;0|0]] ||
|-
| [[subgroup rank of a group]] || [[arithmetic function value::subgroup rank of a group;0|0]] ||
|}
==Group properties==
{| class="wikitable" border="1"
! Property !! Satisfied !! Explanation
|-
| [[satisfies property::cyclic group]] || Yes ||
|-
| [[satisfies property::abelian group]] || Yes ||
|-
| [[satisfies property::nilpotent group]] || Yes ||
|-
| [[satisfies property::solvable group]] || Yes ||
|-
| [[satisfies property::perfect group]] || Yes ||
|}
==Linear representation theory==
{{further|[[Linear representation theory of trivial group]]}}
The trivial group only has one irreducible representation, the trivial representation.
==In Galois theory==
===Galois extensions with trivial Galois group===
{{further|[[Galois extensions for trivial group]]}}
Given a [[Galois extension]] of fields <math>L/K</math>, the Galois group for the extension is the trivial group if and only if <math>L=K</math>.
==Other results==
The [[derived subgroup]] of a group is the trivial group precisely if that group is [[abelian group|abelian]], see the proof [[derived subgroup is trivial if and only if group is abelian|here]].
==GAP implementation==
{{GAP ID|1|1}}
===Other descriptions===
The group can be defined using the [[GAP:TrivialGroup|TrivialGroup]] function:
<tt>TrivialGroup</tt>

Latest revision as of 17:14, 12 January 2024

This article is about a basic definition in group theory. The article text may, however, contain advanced material.
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Definition

Verbal definition

The trivial group is the group with only one element, which is its identity element. The trivial group is usually denoted as , , or .

Alternative definitions

Multiplication table

Element (identity element)

Importance

The trivial group is important in the following ways:

  • For any group, there is a unique homomorphism from the trivial group to that group, namely the homomorphism sending it to the identity element. Thus, the trivial group occurs in a unique way as a subgroup for any given group, namely the one-element subgroup comprising the identity element. This is termed the trivial subgroup.
  • For any group, there is a unique homomorphism to the trivial group from that group, namely the homomorphism sending everything to the identity element. Thus, the trivial group occurs in a unique way as a quotient group of any given group, namely its quotient by itself. This is termed the trivial quotient.

Arithmetic functions

Function Value Explanation
order 1 only the identity element.
exponent 1
nilpotency class 0
derived length 0
Frattini length 0
Fitting length 0
minimum size of generating set 0
subgroup rank of a group 0

Group properties

Property Satisfied Explanation
cyclic group Yes
abelian group Yes
nilpotent group Yes
solvable group Yes
perfect group Yes

Linear representation theory

Further information: Linear representation theory of trivial group

The trivial group only has one irreducible representation, the trivial representation.

In Galois theory

Galois extensions with trivial Galois group

Further information: Galois extensions for trivial group

Given a Galois extension of fields , the Galois group for the extension is the trivial group if and only if .

Other results

The derived subgroup of a group is the trivial group precisely if that group is abelian, see the proof here.

GAP implementation

Group ID

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

SmallGroup(1,1)

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

gap> G := SmallGroup(1,1);

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

IdGroup(G) = [1,1]

or just do:

IdGroup(G)

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


Other descriptions

The group can be defined using the TrivialGroup function:

TrivialGroup

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