group homomorphism

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Image:Group homomorphism ver.2.svg, 250px, 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_(algebra)#Group_homomorphisms, kernel of h and aN is a coset of N. In mathematics, given two group (mathematics), groups, (''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 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 h ...
''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, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homom ...
''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 automata, as well as the computational problem In theoretical computer science An artistic representation of a Turing machine. Turing machines are used to model general computing devices. T ...
, 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 350px, The real numbers form a topological group under addition ">addition.html" ;"title="real numbers form a topological group under addition">real numbers form a topological group under addition In mathematics, a topological group is a group ...
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 220px In the context of abstract algebra or universal algebra, a monomorphism is an Injective function, injective homomorphism. A monomorphism from to is often denoted with the notation X\hookrightarrow Y. In the more general setting of catego ...
: A group homomorphism that is
injective 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). I ...

(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 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). I ...
: A group homomorphism that is
bijective 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 ...

; 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 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). I ...
: A homomorphism, ''h'': ''G'' → ''G''; the domain and codomain are the same. Also called an endomorphism of ''G''. ;
Automorphism 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 ...

: An endomorphism that is bijective, and hence an isomorphism. The set of all
automorphism 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 ...

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 kernel is a computer program at the core of a computer's operating system that has complete control over everything in the system. It is the "portion of the operating system co ...
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 ''
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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
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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
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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 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). ...
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 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). I ...

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
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. 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 theory, category.

# Homomorphisms of abelian groups

If ''G'' and ''H'' are abelian group, 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, this shows that the set End(''G'') of all endomorphisms of an abelian group forms a ring (algebra), ring, the ''endomorphism ring'' of ''G''. For example, the endomorphism ring of the abelian group consisting of the Direct sum of groups, direct sum of ''m'' copies of Z/''n''Z is isomorphic to the ring of ''m''-by-''m'' matrix (mathematics), 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.