Double coset - Revision history

BrianMcMahon: Fixed spelling mistake - "dobule" for "double"

2024-07-23T15:36:55Z

Fixed spelling mistake - "dobule" for "double"

BrianMcMahon: Tidied translations and added German and Spanish (U. Mueller)

2017-11-10T13:54:33Z

Tidied translations and added German and Spanish (U. Mueller)

BrianMcMahon: Style edits to align with printed edition

2017-05-13T13:27:28Z

Style edits to align with printed edition

BrianMcMahon at 13:26, 13 May 2017

2017-05-13T13:26:52Z

MassimoNespolo: lang

2015-03-20T13:36:29Z

lang

BrianMcMahon at 12:09, 6 February 2012

2012-02-06T12:09:43Z

MassimoNespolo at 17:30, 25 April 2007

2007-04-25T17:30:54Z

New page

==Definition==
Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g1 ∈ G and g2 ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements hi ∈ H and kj ∈ K such that:

<div align="center">
g2 = hig1kj
</div>

The [[complex]] Hg1K is called a '''double coset'''

The partition of G into double cosets relative to H and K is a classification, ''i''.''e''. each gi ∈ G belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset Hg1K contains complete left cosets of K and complete right cosets of H.

[[Category:Fundamental crystallography]]

@@ Line 10: / Line 10: @@
 The [[complex]] ''Hg''<sub>1</sub>''K'' is  called a '''double coset'''.
-The partition of ''G'' into double cosets relative to ''H'' and ''K'' is a classification, ''i.e.'' each ''g<sub>i</sub>'' &isin; ''G'' belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset ''Hg''<sub>1</sub>''K'' contains complete left cosets of ''K'' and complete right cosets of ''H''.
+The partition of ''G'' into double cosets relative to ''H'' and ''K'' is a classification, ''i.e.'' each ''g<sub>i</sub>'' &isin; ''G'' belongs to exactly one double coset. It is also a generalization of the [[coset]] decomposition, because the double coset ''Hg''<sub>1</sub>''K'' contains complete left cosets of ''K'' and complete right cosets of ''H''.
 === See also ===

← Older revision		Revision as of 13:54, 10 November 2017
Line 1:		Line 1:
−	<~~Font~~ color="blue">Double coset</~~Font~~> (''Fr''). <~~Font~~ color="black">Doppio coset</~~Font~~> (''It''). <~~Font~~ color="purple">両側剰余類</~~Font~~> (''Ja'').	+	<font color="blue">Double coset</font> (''Fr''). <font color="red">Doppelte Nebenklasse</font> (''Ge''). <font color="black">Doppio coset</font> (''It''). <font color="purple">両側剰余類</font> (''Ja''). <font color="green">Clase lateral doble</font> (''Sp'').

@@ Line 8: / Line 8: @@
 </div>
-The [[complex]] ''Hg''<sub>1</sub>''K'' is  called a '''double coset'''
+The [[complex]] ''Hg''<sub>1</sub>''K'' is  called a '''double coset'''.
 The partition of ''G'' into double cosets relative to ''H'' and ''K'' is a classification, ''i.e.'' each ''g<sub>i</sub>'' &isin; ''G'' belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset ''Hg''<sub>1</sub>''K'' contains complete left cosets of ''K'' and complete right cosets of ''H''.

@@ Line 1: / Line 1: @@
-<Font color="blue">Double coset</Font> (''Fr'').　<Font color="black">Doppio coset</Font> (''It'').　<Font color="purple">両側剰余類</Font> (''Ja'')
+<Font color="blue">Double coset</Font> (''Fr''). <Font color="black">Doppio coset</Font> (''It''). <Font color="purple">両側剰余類</Font> (''Ja'').
-Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> &isin; G and g<sub>2</sub> &isin; G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> &isin; H and k<sub>j</sub> &isin; K such that:
+Let ''G'' be a group, and ''H'' and ''K'' be two [[subgroup]]s of ''G''. One says that the two elements ''g''<sub>1</sub> &isin; ''G'' and ''g''<sub>2</sub> &isin; ''G'' belong to the same '''double coset''' of ''G'' relative to ''H'' and ''K'' if there exist elements ''h<sub>i</sub>'' &isin; ''H'' and ''k<sub>j</sub>'' &isin; ''K'' such that
 <div align="center">
-g<sub>2</sub> = h<sub>i</sub>g<sub>1</sub>k<sub>j</sub>
+''g''<sub>2</sub> = ''h<sub>i</sub>g''<sub>1</sub>''k<sub>j</sub>''.
 </div>
-The [[complex]] Hg<sub>1</sub>K is  called a '''double coset'''
+The [[complex]] ''Hg''<sub>1</sub>''K'' is  called a '''double coset'''
-The partition of G into double cosets relative to H and K is a classification, ''i''.''e''. each g<sub>i</sub> &isin; G belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset Hg<sub>1</sub>K contains complete left cosets of K and complete right cosets of H.
+The partition of ''G'' into double cosets relative to ''H'' and ''K'' is a classification, ''i.e.'' each ''g<sub>i</sub>'' &isin; ''G'' belongs to exactly one dobule coset. It is also a generalization of the [[coset]] decomposition, because the double coset ''Hg''<sub>1</sub>''K'' contains complete left cosets of ''K'' and complete right cosets of ''H''.
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 13:36, 20 March 2015
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		+	<Font color="blue">Double coset</Font> (''Fr'').　<Font color="black">Doppio coset</Font> (''It'').　<Font color="purple">両側剰余類</Font> (''Ja'')
		+
		+
	Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ K such that:		Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ K such that:

← Older revision		Revision as of 12:09, 6 February 2012
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−	~~==Definition==~~
	Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ K such that:		Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ K such that: