In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, a trivial group or zero group is a
group that consists of a single element. All such groups are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between the ...
, so one often speaks of
the trivial group. The single element of the trivial group is the
identity element
In mathematics, an identity element or neutral element of a binary operation is an element that leaves unchanged every element when the operation is applied. For example, 0 is an identity element of the addition of real numbers. This concept is use ...
and so it is usually denoted as such: , , or depending on the context. If the group operation is denoted then it is defined by .
The similarly defined 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
In mathematics, the empty set or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
, which has no elements, hence lacks an identity element, and so cannot be a group.
Definitions
Given any group , the group that consists of only the identity element is a
subgroup
In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G.
Formally, given a group (mathematics), group under a binary operation ...
of , and, being the trivial group, is called the of .
The term, when referred to " has no nontrivial proper subgroups" refers to the only subgroups of being the trivial group and the group itself.
Properties
The trivial group is
cyclic of order ; as such it may be denoted or . If the group operation is called addition, the trivial group is usually denoted by . If the group operation is called multiplication then 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 .
The trivial group serves as the
zero object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
in the
category of groups
In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.
Relation to other categories
The ...
, meaning it is both an
initial object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element) ...
and a
terminal object
In category theory, a branch of mathematics, an initial object of a category is an object in such that for every object in , there exists precisely one morphism .
The dual notion is that of a terminal object (also called terminal element): ...
.
The trivial group can be made a (bi-)
ordered group by equipping it with the trivial
non-strict order .
See also
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References
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Finite groups