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Difference between revisions of "Group isomorphism"

From Online Dictionary of Crystallography

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A '''group isomorphism''' is a special type of [[group homomorphism]]. It is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given 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.
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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 respective [[binary operation]]s in ''G'' and in ''H''. A ''group isomorphism'' from (''G'', *) to (''H'', #) is a [[Mapping|bijection]] from ''G'' to ''H'', ''i''.''e''. a bijective function ''f'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that
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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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The two groups (''G'', *) and (''H'', #) are isomorphic if an isomorphism exists. This is written:
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Two groups (''G'', *) and (''H'', #) are isomorphic if an isomorphism between them exists. This is written:
  
 
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If ''H'' = ''G'' and # = * then the bijection is an [[automorphism]].
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If ''H'' = ''G'' and the binary operations # and * coincide, the bijection is an [[automorphism]].
  
 
[[Category:Fundamental crystallography]]
 
[[Category:Fundamental crystallography]]

Revision as of 14:56, 1 April 2009

Isomorphisme entre groupes (Fr). Gruppenisomorphismus (Ge). Isomorfismo fra gruppi (It). 同形 (Ja).


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 : GH 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.

See also