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Difference between revisions of "Group homomorphism"

From Online Dictionary of Crystallography

(Homorphisms between G and H ≠ G)
(Types of homomorphisms: More details and figures)
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===Homomorphisms mapping elements of different sets (G ≠ H)===
 
===Homomorphisms mapping elements of different sets (G ≠ H)===
  
[[Image:Epimorphism.jpg|thumb|300px|An epimorphism is a surjective homomorphism, ''i''.''e''. a homomorphism where the image spans the whole [[Mapping|codomain]]]]
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[[Image:Epimorphism.jpg|thumb|300px|An '''epimorphism''' is a surjective homomorphism, ''i''.''e''. a homomorphism where the image spans the whole [[Mapping|codomain]]]].
 
An '''epimorphism''' is a [[mapping|surjective]] homomorphism, that is, a homomorphism which is ''onto'' as a function. The image of the homomorphism spans the whole set ''H'': in this case, img( ''f'' ) = ''H''
 
An '''epimorphism''' is a [[mapping|surjective]] homomorphism, that is, a homomorphism which is ''onto'' as a function. The image of the homomorphism spans the whole set ''H'': in this case, img( ''f'' ) = ''H''
  
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[[Image:Monomorphism.jpg|thumb|300px|A '''monomorphism''' is an injective homomorphism, ''i''.''e''. a homomorphism where the the elements of ''H'' are mapped to one and only one element of ''G'']].
 
A '''monomorphism''' is an [[mapping|injective]] homomorphism, that is, a homomorphism which is ''one-to-one'' as a function. In this case, ker( ''f'' ) = {1<sub>''G''</sub> }.
 
A '''monomorphism''' is an [[mapping|injective]] homomorphism, that is, a homomorphism which is ''one-to-one'' as a function. In this case, ker( ''f'' ) = {1<sub>''G''</sub> }.
  
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[[Image:Isomorphism.jpg|thumb|300px|An '''isomorphism''' is a bijective homomorphism, ''i''.''e''. there is a one-to-one correspondence between the elements of ''G'' and those of ''H''. Groups (''G'',*) and (''H'',#) differ only for the notation of their elements]].
 
If the homomorphism ''f'' is a [[mapping|bijection]], then its inverse is also a group homomorphism, and ''f'' is called an '''[[group isomorphism| isomorphism]]'''; the groups (''G'',*) and (''H'',#) are called ''isomorphic'' and differ only in the notation of their elements, while they are identical for all practical purposes.
 
If the homomorphism ''f'' is a [[mapping|bijection]], then its inverse is also a group homomorphism, and ''f'' is called an '''[[group isomorphism| isomorphism]]'''; the groups (''G'',*) and (''H'',#) are called ''isomorphic'' and differ only in the notation of their elements, while they are identical for all practical purposes.
  
 
===Homomorphisms mapping elements of the same set (G = H)===
 
===Homomorphisms mapping elements of the same set (G = H)===
  
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[[Image:Endomorphism.jpg|thumb|300px|An '''endomorphism''' is homomorphism of ''G''  to itself]].
 
An '''endomorphism''' is a homomorphism of a group onto itself: ''f'': ''G'' &rarr; ''G''.
 
An '''endomorphism''' is a homomorphism of a group onto itself: ''f'': ''G'' &rarr; ''G''.
  
A [[mapping|bijective]] (invertible) endomorphism (which is hence an [[group isomorphism| isomorphism]]) is called an '''[[automorphism]]'''. The set of all automorphisms of a group ''G'' forms itself a group, the ''automorphism group'' of ''G'', '''Aut(''G'')'''.  
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A [[mapping|bijective]] (invertible) endomorphism (which is hence an [[group isomorphism| isomorphism]]) is called an '''[[automorphism]]'''. The kernel of the automorphism is the identity of ''G'' (1<sub>G</sub>) and the image of the automorphism coincides with ''G''. The set of all automorphisms of a group (''G'',*) forms itself a group, the ''automorphism group'' of ''G'', '''Aut(''G'')'''.  
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 14:35, 19 March 2009

Homomorphisme de groupes (Fr). Gruppenhomomorphismus (Ge). Homomorfismo de grupos (Sp). Omomorfismo di gruppi (It). 準同形 (Ja).

Groups

Let G and H be two non-empty sets with binary operations * (in G) and # (in H). If * and # are associative in G and H respectively and if G and H contain an identity element and the inverse of each element in them, then (G, *) and (H, #) are two groups.

Homomorphism between groups

A group homomorphism from (G, *) to (H, #) is a function f : GH that preserves the composition law, i.e. such that for all u and v in G:

f(u * v) = f(u) # f(v).


The function f maps the identity element 1G of G to the identity element 1H of H, and it also maps inverses to inverses: f(u-1) = f(u)-1.

Kernel and image

Group homomorphism. The kernel of the homomorphism, ker( f ) is the set of elements of G that are mapped on the identity element of H. The image of the homomorphism, im( f ), is the set of elements of H with one or more corresponding element in G. im( f ) is not required to span the whole H
.

The kernel of the homomorphism is the set of the elements of G that are mapped on the identity of H:

ker( f ) = { u in G : f(u) = 1H }

The image of the homomorphism is the subset of elements of H that are mapped by the homomorphism f:

im( f ) = { f(u) : u in G }.

The kernel is a normal subgroup and the image is a subgroup of H.

Types of homomorphisms

Homomorphisms can be classified according to different criteria, among which the relation between G and H and the nature of the mapping.

Homomorphisms mapping elements of different sets (G ≠ H)

An epimorphism is a surjective homomorphism, i.e. a homomorphism where the image spans the whole codomain
.

An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function. The image of the homomorphism spans the whole set H: in this case, img( f ) = H


A monomorphism is an injective homomorphism, i.e. a homomorphism where the the elements of H are mapped to one and only one element of G
.

A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function. In this case, ker( f ) = {1G }.


An isomorphism is a bijective homomorphism, i.e. there is a one-to-one correspondence between the elements of G and those of H. Groups (G,*) and (H,#) differ only for the notation of their elements
.

If the homomorphism f is a bijection, then its inverse is also a group homomorphism, and f is called an isomorphism; the groups (G,*) and (H,#) are called isomorphic and differ only in the notation of their elements, while they are identical for all practical purposes.

Homomorphisms mapping elements of the same set (G = H)

An endomorphism is homomorphism of G to itself
.

An endomorphism is a homomorphism of a group onto itself: f: GG.


A bijective (invertible) endomorphism (which is hence an isomorphism) is called an automorphism. The kernel of the automorphism is the identity of G (1G) and the image of the automorphism coincides with G. The set of all automorphisms of a group (G,*) forms itself a group, the automorphism group of G, Aut(G).