Template:Short description Template:One source In mathematics, a trivial group or zero group is a group that consists of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usually denoted as such: Template:Tmath, Template:Tmath, or Template:Tmath depending on the context. If the group operation is denoted Template:Tmath then it is defined by Template:Tmath.
The similarly defined Template:Visible anchor 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.
DefinitionsEdit
Given any group Template:Tmath, the group that consists of only the identity element is a subgroup of Template:Tmath, and, being the trivial group, is called the Template:Visible anchor of Template:Tmath.
The term, when referred to "Template:Tmath has no nontrivial proper subgroups" refers to the only subgroups of Template:Tmath being the trivial group Template:Tmath and the group Template:Tmath itself.
PropertiesEdit
The trivial group is cyclic of order Template:Tmath; as such it may be denoted Template:Tmath or Template:Tmath. If the group operation is called addition, the trivial group is usually denoted by Template:Tmath. If the group operation is called multiplication then Template:Tmath 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 Template:Tmath.
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-)ordered group by equipping it with the trivial non-strict order Template:Tmath.
See alsoEdit
ReferencesEdit
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