Difference between revisions of "Group isomorphism"
From Online Dictionary of Crystallography
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| − | < | + | <font color="orange">جماعه التماثل</font> (''Ar''). <font color="blue">Isomorphisme de groupes</font> (''Fr''). <font color="red">Gruppenisomorphismus</font> (''Ge''). <font color="black">Isomorfismo fra gruppi </font> (''It''). <font color="purple">同形</font> (''Ja''). <font color="brown">Изоморфизм групп</font> (''Ru''). <font color="green">Isomorfismo de grupos</font> (''Sp''). |
| − | A '''group isomorphism''' is a special type of [[group homomorphism]]. It is a | + | A '''group isomorphism''' is a special type of [[group homomorphism]]. It is a mapping between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the respective group operations. If there exists an isomorphism between two groups, then the groups are called '''isomorphic'''. Isomorphic groups have the same properties and the same structure of their multiplication table. |
| − | Let (''G'', *) and (''H'', #) be two groups, where | + | Let (''G'', *) and (''H'', #) be two groups, where '*' and '#' are the [[binary operation]]s in ''G'' and ''H'', respectively. A ''group isomorphism'' from (''G'', *) to (''H'', #) is a [[Mapping|bijection]] from ''G'' to ''H'', ''i.e''. a bijective mapping ''f'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' one has |
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| − | + | Two groups (''G'', *) and (''H'', #) are isomorphic if an isomorphism between them exists. This is written: | |
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| − | (''G'', *) <math>\cong</math> (''H'', #) | + | (''G'', *) <math>\cong</math> (''H'', #). |
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| − | If ''H'' = ''G'' and # | + | If ''H'' = ''G'' and the binary operations # and * coincide, the bijection is an [[automorphism]]. |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Latest revision as of 15:10, 29 August 2018
جماعه التماثل (Ar). Isomorphisme de groupes (Fr). Gruppenisomorphismus (Ge). Isomorfismo fra gruppi (It). 同形 (Ja). Изоморфизм групп (Ru). Isomorfismo de grupos (Sp).
A group isomorphism is a special type of group homomorphism. It is a mapping between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the respective group operations. If there exists an isomorphism between two groups, then the groups are called isomorphic. Isomorphic groups have the same properties and the same structure of their multiplication table.
Let (G, *) and (H, #) be two groups, where '*' and '#' are the binary operations in G and H, respectively. A group isomorphism from (G, *) to (H, #) is a bijection from G to H, i.e. a bijective mapping f : G → H such that for all u and v in G one has
f (u * v) = f (u) # f (v).
Two groups (G, *) and (H, #) are isomorphic if an isomorphism between them exists. This is written:
(G, *) [math]\cong[/math] (H, #).
If H = G and the binary operations # and * coincide, the bijection is an automorphism.