View source for Trivial group

{{Short description|Group consisting of only one element}}
In [[Mathematics|mathematics]], a '''trivial group''' or '''zero group''' is a [[Group (mathematics)|group]] consisting 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|identity element]] and so it is usually denoted as such: <math>0, 1,</math> or <math>e</math> depending on the context. If the group operation is denoted <math>\, \cdot \,</math> then it is defined by <math>e \cdot e = e.</math> 

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|empty set]], which has no elements, hence lacks an identity element, and so cannot be a group.

== Definitions ==

Given any group <math>G,</math> the group consisting of only the identity element is a [[Subgroup|subgroup]] of <math>G,</math> and, being the trivial group, is called the '''{{visible anchor|trivial subgroup}}''' of <math>G.</math> 

The term, when referred to "<math>G</math> has no nontrivial proper subgroups" refers to the only subgroups of <math>G</math> being the trivial group <math>\{ e \}</math> and the group <math>G</math> itself.

== Properties ==

The trivial group is [[Cyclic group|cyclic]] of order <math>1</math>; as such it may be denoted <math>\mathrm{Z}_1</math> or <math>\mathrm{C}_1.</math> If the group operation is called addition, the trivial group is usually denoted by <math>0.</math> If the group operation is called multiplication then 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 <math>0 = 1.</math> 

The trivial group serves as the zero object in the [[Category of groups|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 <math>\,\leq.</math>

== 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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