Image of a Group homomorphism(h) from G(left) to H(right). The smaller oval inside H is the image of h. N is the kernel of h and aN is a coset of h. In Mathematics, if G is a group, H is a Subgroup of G, and g is an element of G, then gH

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

h(u * v) = h(u) · h(v)

where the group operation on the left hand side of the equation is that of G and on the right hand side that of H. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Mathematics, given a group G and a Normal subgroup N of G, the quotient group, or factor group, of G In Mathematics, the word kernel has several meanings Kernel may mean a subset associated with a mapping The kernel of a mapping is the set of elements that In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage In Mathematics, especially in the area of Abstract algebra known as Group theory, a semidirect product is a particular way in which a group can In Mathematics, one can often define a direct product of objectsalready known giving a new one The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, a finite group is a group which has finitely many elements Group theory is a mathematical discipline the part of Abstract algebra that studies the Algebraic structures known as groups. In Mathematics, a discrete group is a group G equipped with the Discrete topology. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics and Group theory the term multiplicative group refers to one of the following concepts depending on the context any group \scriptstyle\mathfrak In Mathematics, an additive group may be an Abelian group, when it is written using the symbol + for its Binary operation An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an SIMPLE Group Limited is a conglomeration of separately run companies that each specialised in a particular area of Tax Planning In the history of Mathematics, the origins of Group theory lie in the search for a proof of the general unsolvability of Quintic and higher equations finally Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function

From this property, one can deduce that h maps the identity element e_G of G to the identity element e_H of H, and it also maps inverses to inverses in the sense that h(u^-1) = h(u)^-1. Hence one can say that h "is compatible with the group structure".

Older notations for the homomorphism h(x) may be x_h, though this may be confused as an index or a general subscript. A more recent trend is to write group homomorphisms on the right of their arguments, omitting brackets, so that h(x) becomes simply x h. This approach is especially prevalent in areas of group theory where automata play a role, since it accords better with the convention that automata read words from left to right.

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. In Mathematics, a topological group is a group G together with a topology on G such that the group's binary operation and the

Image and kernel

We define the kernel of h to be

ker(h) = { u in G : h(u) = e_H }

and the image of h to be

im(h) = { h(u) : u in G }.

The kernel is a normal subgroup of G (in fact, h(g^-1 u g) = h(g)^-1 h(u) h(g) = h(g)^-1 e_H h(g) = h(g)^-1 h(g) = e_H) and the image is a subgroup of H. In Mathematics, more specifically in Abstract algebra, a normal subgroup is a special kind of Subgroup. In Group theory, given a group G under a Binary operation * we say that some Subset H of G is a subgroup of The homomorphism h is injective (and called a group monomorphism) if and only if ker(h) = {e_G}.

Examples

• Consider the cyclic group Z/3Z = {0, 1, 2} and the group of integers Z with addition. In Group theory, a cyclic group or monogenous group is a group that can be generated by a single element in the sense that the group has an The map h : Z → Z/3Z with h(u) = u mod 3 is a group homomorphism. In Mathematics, modular arithmetic (sometimes called modulo arithmetic, or clock arithmetic) is a system of Arithmetic for Integers It is surjective and its kernel consists of all integers which are divisible by 3. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every

• 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 exponential function is a function in Mathematics. The application of this function to a value x is written as exp( x) In Mathematics, the real numbers may be described informally in several different ways 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. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted This map is surjective and has the kernel { 2πki : k in Z }, as can be seen from Euler's formula. This article is about Euler's formula in Complex analysis. For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic

• Given any two groups G and H, the map h : G → H which sends every element of G to the identity element of H is a homomorphism; its kernel is all of G.

• Given any group G, the identity map id : G → G with id(u) = u for all u in G is a group homomorphism.

The category of groups

If h : G → H and k : H → K are group homomorphisms, then so is k o h : G → K. This shows that the class of all groups, together with group homomorphisms as morphisms, forms a category. In Set theory and its applications throughout Mathematics, a class is a collection of sets (or sometimes other mathematical objects that can be unambiguously In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets

Types of homomorphic maps

If the homomorphism h is a bijection, then one can show that its inverse is also a group homomorphism, and h is called a group isomorphism; in this case, the groups G and H are called isomorphic: they differ only in the notation of their elements and are identical for all practical purposes. In Mathematics, a bijection, or a bijective function is a function f from a set X to a set Y with the property In Abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in

If h: G → G is a group homomorphism, we call it an endomorphism of G. In Mathematics, an endomorphism is a Morphism (or Homomorphism) from a mathematical object to itself If furthermore it is bijective and hence an isomorphism, it is called an automorphism. In Mathematics, an automorphism is an Isomorphism from a Mathematical object to itself The set of all automorphisms of a group G, with functional composition as operation, forms itself 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/2Z.

An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function. In Mathematics, a function f is said to be surjective or onto, if its values span its whole Codomain; that is for every A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function.

Homomorphisms of abelian groups

If G and H are abelian (i. An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the 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) o f = (h o f) + (k o f)   and    g o (h + k) = (g o h) + (g o k).

This shows that the set End(G) of all endomorphisms of an abelian group forms a ring, the endomorphism ring of G. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real In Abstract algebra, one associates to certain objects a ring, the object's endomorphism ring, which encodes several internal properties of the object For example, the endomorphism ring of the abelian group consisting of the direct sum of two copies of Z/2Z (the Klein four-group) is isomorphic to the ring of 2-by-2 matrices with entries in Z/2Z. The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, the Klein four-group (or just Klein group or Vierergruppe, often symbolized by the letter V) is the group Z2 In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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, specifically in Category theory, a preadditive category is a category that is enriched over the Monoidal category In Mathematics, an abelian category is a category in which Morphisms and objects can be added and in which kernels and Cokernels exist

See also

• Fundamental theorem on homomorphisms

References

• Lang, Serge (2002), Algebra, vol. In Abstract algebra, the Fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects Serge Lang ( May 19, 1927 – September 12, 2005) was a French -born American Mathematician. 211 (3rd ed. ), Graduate Texts in Mathematics, Springer-Verlag .

External links

• Group Homomorphism on PlanetMath