Difference between revisions of "Group homomorphism"
From Online Dictionary of Crystallography
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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]]]] | + | [[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'' | ||
+ | |||
+ | [[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> }. | ||
+ | |||
+ | [[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)=== | ||
+ | [[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'' → ''G''. | An '''endomorphism''' is a homomorphism of a group onto itself: ''f'': ''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 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).
Contents
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 : G → H 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
.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, 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.
Homomorphisms mapping elements of the same set (G = H)
.An endomorphism is a homomorphism of a group onto itself: f: G → G.
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).