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Difference between revisions of "Group homomorphism"

From Online Dictionary of Crystallography

(Image and kernel)
(Image and kernel)
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The ''image of h'' is defined as:
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The ''image of h'' is the subset of elements of ''H'' that are mapped by the homomorphism ''h'':
 
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im(''h'') = { ''h''(''u'') : ''u'' in ''G'' }.
 
im(''h'') = { ''h''(''u'') : ''u'' in ''G'' }.

Revision as of 18:54, 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 that preserves the composition law, i.e. 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 function 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 the homomorphism is the set of the elements of G that are mapped on the identity of H:

ker(h) = { u in G : h(u) = 1H }

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

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 }.