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In mathematics, given two groups, (*G*,∗) and (*H*, ·), a **group homomorphism** from (*G*,∗) to (*H*, ·) is a function *h* : *G* → *H* such that for all *u* and *v* in *G* it holds that

- Intuition
- Types
- Image and kernel
- Examples
- Category of groups
- Homomorphisms of abelian groups
- See also
- References
- External links

where the group operation on the left side of the equation is that of *G* and on the right side that of *H*.

From this property, one can deduce that *h* maps the identity element *e _{G}* of

and it also maps inverses to inverses in the sense that

Hence one can say that *h* "is compatible with the group structure".

In areas of mathematics where one considers groups endowed with additional structure, a *homomorphism* sometimes means a map which respects not only the group structure (as above) but also the extra structure. For example, a homomorphism of topological groups is often required to be continuous.

The purpose of defining a group homomorphism is to create functions that preserve the algebraic structure. An equivalent definition of group homomorphism is: The function *h* : *G* → *H* is a group homomorphism if whenever

*a*∗*b*=*c*we have*h*(*a*) ⋅*h*(*b*) =*h*(*c*).

In other words, the group *H* in some sense has a similar algebraic structure as *G* and the homomorphism *h* preserves that.

- Monomorphism
- A group homomorphism that is injective (or, one-to-one); i.e., preserves distinctness.
- Epimorphism
- A group homomorphism that is surjective (or, onto); i.e., reaches every point in the codomain.
- Isomorphism
- A group homomorphism that is bijective; i.e., injective and surjective. Its inverse is also a group homomorphism. In this case, the groups
*G*and*H*are called*isomorphic*; they differ only in the notation of their elements (except of identity element) and are identical for all practical purposes. I.e. we re-label all elements except identity. - Endomorphism
- A group homomorphism,
*h*:*G*→*G*; the domain and codomain are the same. Also called an endomorphism of*G*. - Automorphism
- A group endomorphism that is bijective, and hence an isomorphism. The set of all automorphisms of a group
*G*, with functional composition as operation, itself forms a group, the*automorphism group*of*G*. It is denoted by Aut(*G*). As an example, the automorphism group of (**Z**, +) contains only two elements, the identity transformation and multiplication with −1; it is isomorphic to (**Z**/2**Z**, +).

We define the * kernel of h* to be the set of elements in *G* which are mapped to the identity in *H*

and the * image of h* to be

The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The first isomorphism theorem states that the image of a group homomorphism, *h*(*G*) is isomorphic to the quotient group *G*/ker *h*.

The kernel of h is a normal subgroup of *G*:

and the image of h is a subgroup of *H*.

The homomorphism, *h*, is a *group monomorphism*; i.e., *h* is injective (one-to-one) if and only if ker(*h*) = {*e*_{G}}. Injection directly gives that there is a unique element in the kernel, and, conversely, a unique element in the kernel gives injection:

- Consider the cyclic group Z
_{3}= (**Z**/3**Z**, +) = ({0, 1, 2}, +) and the group of integers (**Z**, +). The map*h*:**Z**→**Z**/3**Z**with*h*(*u*) =*u*mod 3 is a group homomorphism. It is surjective and its kernel consists of all integers which are divisible by 3.

- The set
forms a group under matrix multiplication. For any complex number

*u*the function*f*:_{u}*G*→**C**defined by^{*} - Consider a multiplicative group of positive real numbers (
**R**^{+}, ⋅) for any complex number*u*. Then the function*f*:_{u}**R**^{+}→**C**defined by

- The exponential map yields a group homomorphism from the group of real numbers
**R**with addition to the group of non-zero real numbers**R*** with multiplication. The kernel is {0} and the image consists of the positive real numbers. - The exponential map also yields a group homomorphism from the group of complex numbers
**C**with addition to the group of non-zero complex numbers**C*** with multiplication. This map is surjective and has the kernel {2π*ki*:*k*∈**Z**}, as can be seen from Euler's formula. Fields like**R**and**C**that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields. - The function , defined by is a homomorphism.
- Consider the two groups and , represented respectively by and , where is the positive real numbers. Then, the function defined by the logarithm function is a homomorphism.

If *h* : *G* → *H* and *k* : *H* → *K* are group homomorphisms, then so is *k* ∘ *h* : *G* → *K*. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category.

If *G* and *H* are abelian (i.e., commutative) groups, then the set Hom(*G*, *H*) of all group homomorphisms from *G* to *H* is itself an abelian group: the sum *h* + *k* of two homomorphisms is defined by

- (
*h*+*k*)(*u*) =*h*(*u*) +*k*(*u*) for all*u*in*G*.

The commutativity of *H* is needed to prove that *h* + *k* is again a group homomorphism.

The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if *f* is in Hom(*K*, *G*), *h*, *k* are elements of Hom(*G*, *H*), and *g* is in Hom(*H*, *L*), then

- (
*h*+*k*) ∘*f*= (*h*∘*f*) + (*k*∘*f*) and*g*∘ (*h*+*k*) = (*g*∘*h*) + (*g*∘*k*).

Since the composition is associative, this shows that the set End(*G*) of all endomorphisms of an abelian group forms a ring, the * endomorphism ring * of *G*. For example, the endomorphism ring of the abelian group consisting of the direct sum of *m* copies of **Z**/*n***Z** is isomorphic to the ring of *m*-by-*m* matrices with entries in **Z**/*n***Z**. The above compatibility also shows that the category of all abelian groups with group homomorphisms forms a preadditive category; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an abelian category.

In mathematics, an **abelian group**, also called a **commutative group**, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is commutative. With addition as an operation, the integers and the real numbers form abelian groups, and the concept of an abelian group may be viewed as a generalization of these examples. Abelian groups are named after early 19th century mathematician Niels Henrik Abel.

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,

In abstract algebra, a **group isomorphism** is a function between two groups that sets up a bijection between the elements of the groups in a way that respects the given group operations. If there exists an isomorphism between two groups, then the groups are called **isomorphic**. From the standpoint of group theory, isomorphic groups have the same properties and need not be distinguished.

In algebra, a **homomorphism** is a structure-preserving map between two algebraic structures of the same type. The word *homomorphism* comes from the Ancient Greek language: ὁμός meaning "same" and μορφή meaning "form" or "shape". However, the word was apparently introduced to mathematics due to a (mis)translation of German *ähnlich* meaning "similar" to ὁμός meaning "same". The term "homomorphism" appeared as early as 1892, when it was attributed to the German mathematician Felix Klein (1849–1925).

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 isomorphism is derived from the Ancient Greek: ἴσος*isos* "equal", and μορφή*morphe* "form" or "shape".

In mathematics, and more specifically in linear algebra, a **linear map** is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. The same names and the same definition are also used for the more general case of modules over a ring; see Module homomorphism.

In mathematics, a **group** is a set with an operation that satisfies the following constraints: the operation is associative and has an identity element, and every element of the set has an inverse element.

In ring theory, a branch of abstract algebra, a **ring homomorphism** is a structure-preserving function between two rings. More explicitly, if *R* and *S* are rings, then a ring homomorphism is a function *f* : *R* → *S* such that *f* is:

In mathematics, specifically abstract algebra, the **isomorphism theorems** are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for groups, rings, vector spaces, modules, Lie algebras, and various other algebraic structures. In universal algebra, the isomorphism theorems can be generalized to the context of algebras and congruences.

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 mathematics, **rings** are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a *ring* is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers. Ring elements may be numbers such as integers or complex numbers, but they may also be non-numerical objects such as polynomials, square matrices, functions, and power series.

In mathematics, the endomorphisms of an abelian group *X* form a ring. This ring is called the **endomorphism ring** of *X*, denoted by End(*X*); the set of all homomorphisms of *X* into itself. Addition of endomorphisms arises naturally in a pointwise manner and multiplication via endomorphism composition. Using these operations, the set of endomorphisms of an abelian group forms a (unital) ring, with the zero map as additive identity and the identity map as multiplicative identity.

An **exact sequence** is a sequence of morphisms between objects such that the image of one morphism equals the kernel of the next.

In group theory, the **quaternion group** Q_{8} (sometimes just denoted by Q) is a non-abelian group of order eight, isomorphic to the eight-element subset of the quaternions under multiplication. It is given by the group presentation

In algebra, a **module homomorphism** is a function between modules that preserves the module structures. Explicitly, if *M* and *N* are left modules over a ring *R*, then a function is called an *R*-*module homomorphism* or an *R*-*linear map* if for any *x*, *y* in *M* and *r* in *R*,

In mathematics, an **algebraic torus**, where a one dimensional torus is typically denoted by , , or , is a type of commutative affine algebraic group commonly found in projective algebraic geometry and toric geometry. Higher dimensional algebraic tori can be modelled as a product of algebraic groups . These groups were named by analogy with the theory of *tori* in Lie group theory. For example, over the complex numbers the algebraic torus is isomorphic to the group scheme , which is the scheme theoretic analogue of the Lie group . In fact, any -action on a complex vector space can be pulled back to a -action from the inclusion as real manifolds.

In mathematics, **Schur's lemma** is an elementary but extremely useful statement in representation theory of groups and algebras. In the group case it says that if *M* and *N* are two finite-dimensional irreducible representations of a group *G* and *φ* is a linear map from *M* to *N* that commutes with the action of the group, then either *φ* is invertible, or *φ* = 0. An important special case occurs when *M* = *N*, i.e. *φ* is a self-map; in particular, any element of the center of a group must act as a scalar operator on *M*. The lemma is named after Issai Schur who used it to prove the Schur orthogonality relations and develop the basics of the representation theory of finite groups. Schur's lemma admits generalisations to Lie groups and Lie algebras, the most common of which are due to Jacques Dixmier and Daniel Quillen.

The representation theory of groups is a part of mathematics which examines how groups act on given structures.

In mathematics, a **complex torus** is a particular kind of complex manifold *M* whose underlying smooth manifold is a torus in the usual sense. Here *N* must be the even number 2*n*, where *n* is the complex dimension of *M*.

In mathematics, **Lie group–Lie algebra correspondence** allows one to correspond a Lie group to a Lie algebra or vice versa, and study the conditions for such a relationship. Lie groups that are isomorphic to each other have Lie algebras that are isomorphic to each other, but the converse is not necessarily true. One obvious counterexample is and which are non-isomorphic to each other as Lie groups but their Lie algebras are isomorphic to each other. However, for simply connected Lie groups, the Lie group-Lie algebra correspondence is one-to-one.

- Dummit, D. S.; Foote, R. (2004).
*Abstract Algebra*(3rd ed.). Wiley. pp. 71–72. ISBN 978-0-471-43334-7. - Lang, Serge (2002),
*Algebra*, Graduate Texts in Mathematics, vol. 211 (Revised third ed.), New York: Springer-Verlag, ISBN 978-0-387-95385-4, MR 1878556, Zbl 0984.00001

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

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