Trivial group: Difference between revisions

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


{{basicdef}}
<section begin=beginner/>
 
==Definition==
==Definition==


===Verbal definition===
===Verbal definition===


The '''trivial group''' is the group with only one element, which is its identity element.
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
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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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This article is about a particular group, i.e., a group unique upto isomorphism. View specific information (such as linear representation theory, subgroup structure) about this group
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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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