Difference between revisions of "Group homomorphism"
From Online Dictionary of Crystallography
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==Homomorphism between groups== | ==Homomorphism between groups== | ||
| − | A '''group homomorphism''' from (''G'', *) to (''H'', #) is a function '' | + | 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'': |
<div align="center"> | <div align="center"> | ||
| − | '' | + | ''f''(''u'' * ''v'') = ''f''(''u'') # ''f''(''v'') |
</div> | </div> | ||
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''. | 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 '' | + | The function ''f'' maps the identity element 1<sub>''G''</sub> of ''G'' to the identity element 1<sub>''H''</sub> of ''H'', and it also maps inverses to inverses: ''f''(''u''<sup>-1</sup>) = ''f''(''u'')<sup>-1</sup>. |
== Image and kernel == | == Image and kernel == | ||
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The ''kernel'' of the homomorphism is the set of the elements of ''G'' that are [[mapping|mapped]] on the identity of ''H'': | The ''kernel'' of the homomorphism is the set of the elements of ''G'' that are [[mapping|mapped]] on the identity of ''H'': | ||
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| − | ker('' | + | ker( ''f'' ) = { ''u'' in ''G'' : ''f''(''u'') = 1<sub>''H''</sub> } |
</div> | </div> | ||
| − | The ''image | + | The ''image'' of the homomorphism is the subset of elements of ''H'' that are mapped by the homomorphism ''f'': |
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| − | im('' | + | im( ''f'' ) = { ''f''(''u'') : ''u'' in ''G'' }. |
</div> | </div> | ||
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== Types of homomorphisms == | == Types of homomorphisms == | ||
| − | *If the homomorphism '' | + | *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: '' | + | *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 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('' | + | *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('' | + | *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 13:50, 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 : 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)
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 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
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
- 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: 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: 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 }.