Difference between revisions of "Group homomorphism"
From Online Dictionary of Crystallography
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*An '''endomorphism''' is a homomorphism of a group onto itself: ''h'': ''G'' → ''G''. | *An '''endomorphism''' is a homomorphism of a group onto itself: ''h'': ''G'' → ''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. | + | *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'': img''h'') = ''H'' |
*A '''monomorphism''' is an [[mapping|injective]] homomorphism, that is, a homomorphism which is ''one-to-one'' as a function. In this case, ker(''h'') = {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(''h'') = {1<sub>''G''</sub> }. | ||
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 19:06, 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 : G → H 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: G → G.
- 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. The image of the homomorphism spans the whole set H: imgh) = H
- A monomorphism is an injective homomorphism, that is, a homomorphism which is one-to-one as a function. In this case, ker(h) = {1G }.