In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a trivial group or zero group is a
group consisting of a single element. All such groups are
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping 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 them. The word i ...
, 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:
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 is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
, which has no elements, hence lacks an identity element, and so cannot be a group.
Definitions
Given any group
the group consisting of only the identity element is a
subgroup 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
Cycle, cycles, or cyclic may refer to:
Anthropology and social sciences
* Cyclic history, a theory of history
* Cyclical theory, a theory of American political history associated with Arthur Schlesinger, Sr.
* Social cycle, various cycles in so ...
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 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
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
See also
*
*
References
*
{{DEFAULTSORT:Trivial Group
Finite groups