TheInfoList

In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, given two
groups A group is a number of people or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic identi ...
, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a
function Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...
''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that :$h\left(u*v\right) = h\left(u\right) \cdot h\left(v\right)$ 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 In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...
''eG'' of ''G'' to the identity element ''eH'' of ''H'', :$h\left(e_G\right) = e_H$ and it also maps inverses to inverses in the sense that :$h\left\left(u^\right\right) = h\left(u\right)^. \,$ Hence one can say that ''h'' "is compatible with the group structure". Older notations for the
homomorphism In algebra Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...
''h''(''x'') may be ''x''''h'' or ''x''''h'', though this may be confused as an index or a general subscript. In
automata theory Automata theory is the study of abstract machines and automaton, automata, as well as the computational problems that can be solved using them. It is a theory in theoretical computer science. The word ''automata'' (the plural of ''automaton'') com ...

, sometimes homomorphisms are written to the right of their arguments without parentheses, so that ''h''(''x'') becomes simply ''x h''. 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 group In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...
s is often required to be continuous.

# Intuition

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.

# Types

;
Monomorphism In the context of abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, rin ...
: A group homomorphism that is
injective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

(or, one-to-one); i.e., preserves distinctness. ;
Epimorphism 220px In category theory Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labe ...
: A group homomorphism that is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

(or, onto); i.e., reaches every point in the codomain. ;
Isomorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...
: A group homomorphism that is
bijective In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

; 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 and are identical for all practical purposes. ;
Endomorphism In mathematics, an endomorphism is a morphism from a mathematical object to itself. An endomorphism that is also an isomorphism is an automorphism. For example, an endomorphism of a vector space is a linear map , and an endomorphism of a group ...
: A homomorphism, ''h'': ''G'' → ''G''; the domain and codomain are the same. Also called an endomorphism of ''G''. ;
Automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

: An endomorphism that is bijective, and hence an isomorphism. The set of all
automorphism In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s 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.

# Image and kernel

We define the ''
kernel Kernel may refer to: Computing * Kernel (operating system), the central component of most operating systems * Kernel (image processing), a matrix used for image convolution * Compute kernel, in GPGPU programming * Kernel method, in machine learnin ...
of h'' to be the set of elements in ''G'' which are mapped to the identity in ''H'' : $\operatorname\left(h\right) \equiv \left\.$ and the ''
image An image (from la, imago) is an artifact that depicts visual perception Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...
of h'' to be : $\operatorname\left(h\right) \equiv h\left(G\right) \equiv \left\.$ The kernel and image of a homomorphism can be interpreted as measuring how close it is to being an isomorphism. The
first isomorphism theorem In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...
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 In abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...
of ''G'' and the image of h is a
subgroup In group theory, a branch of mathematics, given a group (mathematics), 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 ...
of ''H'': : $\begin h\left\left(g^ \circ u \circ g\right\right) &= h\left(g\right)^ \cdot h\left(u\right) \cdot h\left(g\right) \\ &= h\left(g\right)^ \cdot e_H \cdot h\left(g\right) \\ &= h\left(g\right)^ \cdot h\left(g\right) = e_H. \end$ If and only if , the homomorphism, ''h'', is a ''group monomorphism''; i.e., ''h'' is injective (one-to-one). Injection directly gives that there is a unique element in the kernel, and a unique element in the kernel gives injection: :$\begin && h\left(g_1\right) &= h\left(g_2\right) \\ \Leftrightarrow && h\left(g_1\right) \cdot h\left(g_2\right)^ &= e_H \\ \Leftrightarrow && h\left\left(g_1 \circ g_2^\right\right) &= e_H,\ \operatorname\left(h\right) = \ \\ \Rightarrow && g_1 \circ g_2^ &= e_G \\ \Leftrightarrow && g_1 &= g_2 \end$

# Examples

* Consider the
cyclic group In group theory The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

Z/3Z = and the group of integers Z with addition. The map ''h'' : Z → Z/3Z with ''h''(''u'') = ''u'' mod 3 is a group homomorphism. It is
surjective In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...
and its kernel consists of all integers which are divisible by 3. * The yields a group homomorphism from the group of
real number In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...
s R with addition to the group of non-zero real numbers R* with multiplication. The kernel is and the image consists of the positive real numbers. * The exponential map also yields a group homomorphism from the group of
complex number In mathematics, a complex number is an element of a number system that contains the real numbers and a specific element denoted , called the imaginary unit, and satisfying the equation . Moreover, every complex number can be expressed in the for ...

s C with addition to the group of non-zero complex numbers C* with multiplication. This map is surjective and has the kernel , as can be seen from
Euler's formula Euler's formula, named after Leonhard Euler Leonhard Euler ( ; ; 15 April 170718 September 1783) was a Swiss mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) incl ...

. Fields like R and C that have homomorphisms from their additive group to their multiplicative group are thus called exponential fields.

# The category of groups

If and are group homomorphisms, then so is . This shows that the class of all groups, together with group homomorphisms as morphisms, forms a
category Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity to recognize shared features or similarities between the elements of the experience of the world (such as O ...
.

# Homomorphisms of abelian groups

If ''G'' and ''H'' are abelian (i.e., commutative) groups, then the set of all group homomorphisms from ''G'' to ''H'' is itself an abelian group: the sum 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 is again a group homomorphism. The addition of homomorphisms is compatible with the composition of homomorphisms in the following sense: if ''f'' is in , ''h'', ''k'' are elements of , and ''g'' is in , then :    and    . Since the composition is
associative In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...
, this shows that the set End(''G'') of all endomorphisms of an abelian group forms a ring, the ''
endomorphism ringIn abstract algebra, the endomorphisms of an abelian group In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometr ...
'' of ''G''. For example, the endomorphism ring of the abelian group consisting of the
direct sum The direct sum is an operation from abstract algebra In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...
of ''m'' copies of Z/''n''Z is isomorphic to the ring of ''m''-by-''m''
matrices Matrix or MATRIX may refer to: Science and mathematics * Matrix (mathematics) In , a matrix (plural matrices) is a array or table of s, s, or s, arranged in rows and columns, which is used to represent a or a property of such an object. Fo ...
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 In mathematics, specifically in category theory, a preadditive category is another name for an Ab-category, i.e., a category (mathematics), category that is enriched category, enriched over the category of abelian groups, Ab. That is, an Ab-catego ...
; the existence of direct sums and well-behaved kernels makes this category the prototypical example of an
abelian category In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mo ...
.