Group isomorphism - Revision history

MassimoNespolo at 15:10, 29 August 2018

2018-08-29T15:10:54Z

MassimoNespolo: Lang (Ar)

2018-08-29T14:51:17Z

Lang (Ar)

MassimoNespolo: Lang (Ru)

2018-08-29T12:19:01Z

Lang (Ru)

BrianMcMahon: Added Spanish translation (U. Mueller)

2017-11-13T14:46:03Z

Added Spanish translation (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-15T12:32:22Z

Style edits to align with printed edition

BerndSouvignier at 14:56, 1 April 2009

2009-04-01T14:56:18Z

MassimoNespolo: link

2009-03-19T14:39:24Z

link

MassimoNespolo: lin

2008-12-21T18:32:59Z

lin

MassimoNespolo at 09:01, 4 July 2007

2007-07-04T09:01:14Z

MassimoNespolo at 10:16, 29 May 2007

2007-05-29T10:16:40Z

← Older revision		Revision as of 15:10, 29 August 2018
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−	<font color="orange">جماعه ~~التماث~~</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'').	+	<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'').

← Older revision		Revision as of 14:51, 29 August 2018
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−	<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'').	+	<font color="orange">جماعه التماث</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'').

← Older revision		Revision as of 12:19, 29 August 2018
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−	<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="green">Isomorfismo de grupos</font> (''Sp'').	+	<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'').

← Older revision		Revision as of 14:46, 13 November 2017
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−	<~~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="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="green">Isomorfismo de grupos</font> (''Sp'').

@@ Line 1: / Line 1: @@
-<Font color="blue">Isomorphisme entre 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="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'').
 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 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
+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
 <div align="center">
@@ Line 13: / Line 13: @@
 <div align="center">
-(''G'', *) <math>\cong</math> (''H'', #)
+(''G'', *) <math>\cong</math> (''H'', #).
 </div>

@@ Line 2: / Line 2: @@
-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.
+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
+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
 <div align="center">
@@ Line 10: / Line 10: @@
 </div>
-The two groups (''G'', *) and (''H'', #) are isomorphic if an isomorphism exists. This is written:
+Two groups (''G'', *) and (''H'', #) are isomorphic if an isomorphism between them exists. This is written:
 <div align="center">
@@ Line 16: / Line 16: @@
 </div>
-If ''H'' = ''G'' and # = * then the bijection is an [[automorphism]].
+If ''H'' = ''G'' and the binary operations # and * coincide, the bijection is an [[automorphism]].
 [[Category:Fundamental 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.
-Let (''G'', *) and (''H'', #) be two groups, where "*" and "#" are the respective binary operations in ''G'' and in ''H''. A ''group isomorphism'' from (''G'', *) to (''H'', #) is a 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
+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 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
 <div align="center">

← Older revision		Revision as of 09:01, 4 July 2007
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	[[Category:Fundamental crystallography]]		[[Category:Fundamental crystallography]]
		+
		+	==See also==
		+	* [[Isomorphous crystals]]

← Older revision		Revision as of 10:16, 29 May 2007
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−	A '''group isomorphism''' 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.	+	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.

	Let (''G'', ) and (''H'', #) be two groups, where "" and "#" are the respective binary operations in ''G'' and in ''H''. A ''group isomorphism'' from (''G'', *) to (''H'', #) is a 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		Let (''G'', ) and (''H'', #) be two groups, where "" and "#" are the respective binary operations in ''G'' and in ''H''. A ''group isomorphism'' from (''G'', *) to (''H'', #) is a 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