Editing Trivial group

{{Short description|Group that has only one element}}
{{one source |date=May 2024}}
In [[mathematics]], a '''trivial group''' or '''zero group''' is a [[Group (mathematics)|group]] that consists of a single element. All such groups are [[isomorphic]], so one often speaks of <em>the</em> trivial group. The single element of the trivial group is the [[identity element]] and so it is usually denoted as such: {{tmath|1= 0 }}, {{tmath|1= 1 }}, or {{tmath|1= \mathrm{e} }} depending on the context. If the group operation is denoted {{tmath|1= \, \cdot \, }} then it is defined by {{tmath|1= \mathrm{e} \cdot \mathrm{e} = \mathrm{e} }}. 

The similarly defined '''{{visible anchor|trivial monoid}}''' is also a group since its only element is its own inverse, and is hence the same as the trivial group. 

The trivial group is distinct from the [[empty set]], which has no elements, hence lacks an identity element, and so cannot be a group.

== Definitions ==
Given any group {{tmath|1= G }}, the group that consists of only the identity element is a [[subgroup]] of {{tmath|1= G }}, and, being the trivial group, is called the '''{{visible anchor|trivial subgroup}}''' of {{tmath|1= G }}.

The term, when referred to "{{tmath|1= G }} has no nontrivial proper subgroups" refers to the only subgroups of {{tmath|1= G }} being the trivial group {{tmath|1= \{ \mathrm{e} \} }} and the group {{tmath|1= G }} itself.

== Properties ==
The trivial group is [[Cyclic group|cyclic]] of order {{tmath|1= 1 }}; as such it may be denoted {{tmath|1= \mathrm{Z}_1 }} or {{tmath|1= \mathrm{C}_1 }}. If the group operation is called addition, the trivial group is usually denoted by {{tmath|1= 0 }}. If the group operation is called multiplication then {{tmath|1= 1 }} can be a notation for the trivial group. Combining these leads to the [[trivial ring]] in which the addition and multiplication operations are identical and {{tmath|1= 0 = 1 }}. 

The trivial group serves as the [[zero object]] in the [[category of groups]], meaning it is both an [[initial object]] and a [[terminal object]].

The trivial group can be made a (bi-)[[Linearly ordered group|ordered group]] by equipping it with the trivial [[non-strict order]] {{tmath|1= \,\leq }}.

== See also ==
* {{annotated link|Zero object (algebra)}}
* {{annotated link|List of small groups}}

== References ==
{{reflist}}
* {{MathWorld |title=Trivial Group |urlname=TrivialGroup |author=Rowland, Todd |author2=Weisstein, Eric W. |name-list-style=amp }}

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[[Category:Finite groups]]