# Trivial group

Last updated

In mathematics, a trivial group or zero 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 and so it is usually denoted as such: ${\displaystyle 0,1,}$ or ${\displaystyle e}$ depending on the context. If the group operation is denoted ${\displaystyle \,\cdot \,}$ then it is defined by ${\displaystyle e\cdot e=e.}$

## Contents

The similarly defined 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, which has no elements, hence lacks an identity element, and so cannot be a group.

## Definitions

Given any group ${\displaystyle G,}$ the group consisting of only the identity element is a subgroup of ${\displaystyle G,}$ and, being the trivial group, is called the trivial subgroup of ${\displaystyle G.}$

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

## Properties

The trivial group is cyclic of order ${\displaystyle 1}$; as such it may be denoted ${\displaystyle Z_{1}}$ or ${\displaystyle C_{1}.}$ If the group operation is called addition, the trivial group is usually denoted by ${\displaystyle 0.}$ 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 ${\displaystyle 0=1.}$

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 ${\displaystyle \,\leq .}$

## Related Research Articles

In abstract algebra, the center of a group, G, is the set of elements that commute with every element of G. It is denoted Z(G), from German Zentrum, meaning center. In set-builder notation,

A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure. For example, the cyclic group of addition modulo n can be obtained from the group of integers under addition by identifying elements that differ by a multiple of n and defining a group structure that operates on each such class as a single entity. It is part of the mathematical field known as group theory.

In mathematics, given two groups, and, a group homomorphism from to is a function h : GH such that for all u and v in G it holds that

In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism group of the structure. It is said that the group acts on the space or structure. If a group acts on a structure, it will usually also act on objects built from that structure. For example, the group of Euclidean isometries acts on Euclidean space and also on the figures drawn in it. In particular, it acts on the set of all triangles. Similarly, the group of symmetries of a polyhedron acts on the vertices, the edges, and the faces of the polyhedron.

In mathematics, a group is a set equipped with an operation that combines any two elements of the set to produce a third element of the set, in such a way that the operation is associative, an identity element exists and every element has an inverse. These three conditions, called group axioms, hold for number systems and many other mathematical structures. For example, the integers together with the addition operation form a group. The concept of a group and its definition through the group axioms was elaborated for handling, in a unified way, essential structural properties of entities of very different mathematical nature. Because of the ubiquity of groups in numerous areas, some authors consider them as a central organizing principle of contemporary mathematics.

In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation.

In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product:

In group theory, Cayley's theorem, named in honour of Arthur Cayley, states that every group G is isomorphic to a subgroup of a symmetric group. More specifically, G is isomorphic to a subgroup of the symmetric group whose elements are the permutations of the underlying set of G. Explicitly,

In mathematics, a free abelian group is an abelian group with a basis. Being an abelian group means that it is a set with an addition operation that is associative, commutative, and invertible. A basis, also called an integral basis, is a subset such that every element of the group can be uniquely expressed as an integer combination of finitely many basis elements. For instance the two-dimensional integer lattice forms a free abelian group, with coordinatewise addition as its operation, and with the two points (1,0) and (0,1) as its basis. Free abelian groups have properties which make them similar to vector spaces, and may equivalently be called free-modules, the free modules over the integers. Lattice theory studies free abelian subgroups of real vector spaces. In algebraic topology, free abelian groups are used to define chain groups, and in algebraic geometry they are used to define divisors.

A group is a set together with an associative operation which admits an identity element and such that every element has an inverse.

Ring theory is the branch of mathematics in which rings are studied: that is, structures supporting both an addition and a multiplication operation. This is a glossary of some terms of the subject.

In mathematics, a module is a generalization of the notion of vector space, wherein the field of scalars is replaced by a ring. The concept of module is also a generalization of the one of abelian group, since the abelian groups are exactly the modules over the ring of integers.

In algebra, a valuation is a function on a field that provides a measure of size or multiplicity of elements of the field. It generalizes to commutative algebra the notion of size inherent in consideration of the degree of a pole or multiplicity of a zero in complex analysis, the degree of divisibility of a number by a prime number in number theory, and the geometrical concept of contact between two algebraic or analytic varieties in algebraic geometry. A field with a valuation on it is called a valued field.

In algebra, a group ring is a free module and at the same time a ring, constructed in a natural way from any given ring and any given group. As a free module, its ring of scalars is the given ring, and its basis is the set of elements of the given group. As a ring, its addition law is that of the free module and its multiplication extends "by linearity" the given group law on the basis. Less formally, a group ring is a generalization of a given group, by attaching to each element of the group a "weighting factor" from a given ring.

In mathematics, a principal homogeneous space, or torsor, for a group G is a homogeneous space X for G in which the stabilizer subgroup of every point is trivial. Equivalently, a principal homogeneous space for a group G is a non-empty set X on which G acts freely and transitively . An analogous definition holds in other categories, where, for example,

In abstract algebra, a semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse.

In mathematics, a zero element is one of several generalizations of the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context.

In mathematics, specifically in group theory, the direct product is an operation that takes two groups G and H and constructs a new group, usually denoted G × H. This operation is the group-theoretic analogue of the Cartesian product of sets and is one of several important notions of direct product in mathematics.

In mathematics, the additive identity of a set that is equipped with the operation of addition is an element which, when added to any element x in the set, yields x. One of the most familiar additive identities is the number 0 from elementary mathematics, but additive identities occur in other mathematical structures where addition is defined, such as in groups and rings.

In algebra, the zero object of a given algebraic structure is, in the sense explained below, the simplest object of such structure. As a set it is a singleton, and as a magma has a trivial structure, which is also an abelian group. The aforementioned abelian group structure is usually identified as addition, and the only element is called zero, so the object itself is typically denoted as {0}. One often refers to the trivial object since every trivial object is isomorphic to any other.

## References

• Rowland, Todd & Weisstein, Eric W. "Trivial Group". MathWorld .