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: or depending on the context. If the group operation is denoted then it is defined by

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.

Given any group the group consisting of only the identity element is a subgroup of and, being the trivial group, is called the **trivial subgroup** 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.

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 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

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* : *G* → *H* 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.

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

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.