View source for Trivial group

{{basicdef}}
{{particular 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 <math>1</math>, <math>\{ 1 \}</math>, or <math>\{ e \}</math>.
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===Alternative definitions===

* The {{cyclic group}} on one element
* The {{symmetric group}} on one element
* The {{projective general 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==

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'''.
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==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 ||
|}
==GAP implementation==

{{GAP ID|1|1}}

===Other descriptions===

The group can be defined using the [[GAP:TrivialGroup|TrivialGroup]] function:

<tt>TrivialGroup</tt>