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Difference between revisions of "Group homomorphism"

From Online Dictionary of Crystallography

(Image and kernel: figure caption)
(Types of homomorphisms)
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== Types of homomorphisms ==
 
== Types of homomorphisms ==
  
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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]]]]
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*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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*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> }.
 
*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.
 
 
*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'')'''.  
 
*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'')'''.  
*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''
 
*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> }.
 
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 14:02, 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.

Image and kernel

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

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, that is, a homomorphism which is one-to-one as a function. In this case, ker( f ) = {1G }.
  • 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.
  • 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 set of all automorphisms of a group G forms itself a group, the automorphism group of G, Aut(G).