Trivial group
Encyclopedia
In mathematics
, a trivial group is a group
consisting 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
so it usually denoted as such, 0, 1 or e depending on the context. The trivial group is often written as 0, 1, or e, using the same notation for the group and its sole element since there is little danger of confusion. If the group operation is denoted * then it is defined by e * e = e.
The trivial group should not be confused with the empty set
(which has no elements and therefore, lacking an identity element, cannot be a group).
Given an arbitrary group G, the group consisting of just the identity element is a trivial group and is a subgroup
of G, it is called the trivial subgroup.
of order 1; as such it may be denoted Z1.
The trivial group serves as the zero object in the category of groups
, meaning it is both an initial object
and a terminal object.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, a trivial group is a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
consisting 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
Identity element
In mathematics, an identity element is a special type of element of a set with respect to a binary operation on that set. It leaves other elements unchanged when combined with them...
so it usually denoted as such, 0, 1 or e depending on the context. The trivial group is often written as 0, 1, or e, using the same notation for the group and its sole element since there is little danger of confusion. If the group operation is denoted * then it is defined by e * e = e.
The trivial group should not be confused with the empty set
Empty set
In mathematics, and more specifically set theory, the empty set is the unique set having no elements; its size or cardinality is zero. Some axiomatic set theories assure that the empty set exists by including an axiom of empty set; in other theories, its existence can be deduced...
(which has no elements and therefore, lacking an identity element, cannot be a group).
Given an arbitrary group G, the group consisting of just the identity element is a trivial group and is a subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
of G, it is called the trivial subgroup.
Properties
The trivial group is the cyclicCyclic group
In group theory, a cyclic group is a group that can be generated by a single element, in the sense that the group has an element g such that, when written multiplicatively, every element of the group is a power of g .-Definition:A group G is called cyclic if there exists an element g...
of order 1; as such it may be denoted Z1.
The trivial group serves as the zero object in the category of groups
Category of groups
In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category...
, meaning it is both an initial object
Initial object
In category theory, an abstract branch of mathematics, an initial object of a category C is an object I in C such that for every object X in C, there exists precisely one morphism I → X...
and a terminal object.