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Difference between revisions of "Group homomorphism"

From Online Dictionary of Crystallography

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<font color="blue">Homomorphisme de groupes </font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="green">Homomorfismo de grupos </font> (''Sp''). <font color="black">Omomorfismo di gruppi </font> (''It''). <Font color="purple">準同形</font> (''Ja'').
 
<font color="blue">Homomorphisme de groupes </font> (''Fr''). <font color="red">Gruppenhomomorphismus</font> (''Ge''). <font color="green">Homomorfismo de grupos </font> (''Sp''). <font color="black">Omomorfismo di gruppi </font> (''It''). <Font color="purple">準同形</font> (''Ja'').
  
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==Groups==
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Let ''G'' and ''H'' be two non-empty sets with [[binary operation]]s * (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.
  
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==Homomorphism between groups==
 
A '''group homomorphism''' from (''G'', *) to (''H'', ·) is a function ''h'' : ''G'' &rarr; ''H'' such that for all ''u'' and ''v'' in ''G'':
 
A '''group homomorphism''' from (''G'', *) to (''H'', ·) is a function ''h'' : ''G'' &rarr; ''H'' such that for all ''u'' and ''v'' in ''G'':
 
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Revision as of 18:46, 18 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 h : GH such that for all u and v in G:

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.

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

Image and kernel

The kernel of h is defined as:

ker(h) = { u in G : h(u) = 1H }

in other words, the kernel of the homomorphism is the set of the elements of G that are mapped on the identity of H.

The image of h is defined as:

im(h) = { h(u) : u in G }.

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

Types of homomorphisms

  • If the homomorphism h is a bijection, then its inverse is also a group homomorphism, and h 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: h: 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).
  • An epimorphism is a surjective homomorphism, that is, a homomorphism which is onto as a function.
  • A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function. In this case, ker(h) = {1G }.