Coset - Revision history

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

2017-11-09T17:20:49Z

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

MassimoNespolo: two languages added

2017-10-10T15:26:41Z

two languages added

BrianMcMahon: Style edits to align with printed edition

2017-05-13T10:53:21Z

Style edits to align with printed edition

MassimoNespolo at 17:54, 25 April 2007

2007-04-25T17:54:46Z

MassimoNespolo at 17:31, 25 April 2007

2007-04-25T17:31:46Z

MassimoNespolo at 17:31, 25 April 2007

2007-04-25T17:31:24Z

MassimoNespolo at 17:16, 25 April 2007

2007-04-25T17:16:19Z

MassimoNespolo at 09:37, 26 February 2007

2007-02-26T09:37:45Z

New page

==Definition==
If G is a grup, H a of subgroup G, and g an element of G, then
:gH = { gh : h ∈ H } is a '''left coset of H''' in ''G''
:Hg = { hg : h ∈ H } is a '''right coset of H''' in ''G''.

The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless H is a [[normal subgroup]] of G.

Any two left cosets are either identical or disjoint: the left cosets form a partition of G, because every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is H itself; this is also the only coset that is a subgroup. The same holds for right cosets.

All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of H, because H is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of H in G, written as [G : H] and given by Lagrange's theorem:
:|G|/|H| = [G : H].

== Example ==
The coset decomposition of the [[twin lattice]] point group with respect to the point group of the individual gives the different possible [[twin law]]s. Each element in a coset is a possible [[twin operation]].

[[Category:Fundamental crystallography]]

← Older revision		Revision as of 17:20, 9 November 2017
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−	<font color="orange">مجموعة مشاركة</font> (''Ar''); <font color="blue">Co-ensemble</font> (''Fr''); <font color="red">Restklasse</font> (''Ge''); <font color="black">Classe laterale</font> (''It''); <font color="purple">剰余類</font> (''Ja''); <font color="green">Clase lateral</font> (''Sp'').	+	<font color="orange">مجموعة مشاركة</font> (''Ar''). <font color="blue">Co-ensemble</font> (''Fr''). <font color="red">Nebenklasse, Restklasse</font> (''Ge''). <font color="black">Classe laterale</font> (''It''). <font color="purple">剰余類</font> (''Ja''). <font color="green">Clase lateral, clase adjunta</font> (''Sp'').

	==Definition==		==Definition==

← Older revision		Revision as of 15:26, 10 October 2017
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−	<font color="blue">Co-ensemble</font> (''Fr''). <font color="red">Restklasse</font> (''Ge''). <font color="black">Classe laterale</font> (''It''). <font color="purple">剰余類</font> (''Ja'').	+	<font color="orange">مجموعة مشاركة</font> (''Ar''); <font color="blue">Co-ensemble</font> (''Fr''); <font color="red">Restklasse</font> (''Ge''); <font color="black">Classe laterale</font> (''It''); <font color="purple">剰余類</font> (''Ja''); <font color="green">Clase lateral</font> (''Sp'').

	==Definition==		==Definition==

@@ Line 1: / Line 1: @@
-<font color="blue">Classe suivant un sous-groupe</font> (''Fr''). <font color="red">Restklasse</font> (''Ge''). <font color="black">Classe laterale</font> (''It''). <font color="purple">剰余類</font> (''Ja'').
+<font color="blue">Co-ensemble</font> (''Fr''). <font color="red">Restklasse</font> (''Ge''). <font color="black">Classe laterale</font> (''It''). <font color="purple">剰余類</font> (''Ja'').
 ==Definition==
-If G is a group, H a [[subgroup]] of G, and g an element of G, then
+If ''G'' is a group, ''H'' a [[subgroup]] of ''G'', and ''g'' an element of ''G'', then
-:gH = { gh : h &isin; H } is a '''left coset of H''' in ''G''
+:''gH'' = { ''gh'' : ''h'' &isin; ''H'' } is a '''left coset of ''H'' ''' in ''G'',
-:Hg = { hg : h &isin; H } is a '''right coset of H''' in ''G''.
+:''Hg'' = { ''hg'' : ''h'' &isin; ''H'' } is a '''right coset of ''H'' ''' in ''G''.
-The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless H is a [[normal subgroup]] of G.
+The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless ''H'' is a [[normal subgroup]] of ''G''.
-Any two left cosets are either identical or disjoint: the left cosets form a partition of G, because every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is H itself; this is also the only coset that is a subgroup. The same holds for right cosets.
+Any two left cosets are either identical or disjoint: the left cosets form a partition of ''G'', because every element of ''G'' belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is ''H'' itself; this is also the only coset that is a subgroup. The same holds for right cosets.
-All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of H, because H is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of H in G, written as [G : H] and given by Lagrange's theorem:
+All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of ''H'', because ''H'' is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the '''index''' of ''H'' in ''G'', written as [''G'' : ''H''] and given by Lagrange's theorem:
-:|G|/|H| = [G : H].
+:|''G''|/|''H''| = [''G'' : ''H''].
 Cosets are also sometimes called ''associate [[complex]]es''.
@@ Line 19: / Line 19: @@
 ==See also==
-[[Double coset]]
+*[[Double coset]]
 [[Category:Fundamental crystallography]]

@@ Line 2: / Line 2: @@
 ==Definition==
-If G is a group, H a of [[subgroup]] G, and g an element of G, then
+If G is a group, H a [[subgroup]] of G, and g an element of G, then
 :gH = { gh : h &isin; H } is a '''left coset of H''' in ''G''
 :Hg = { hg : h &isin; H } is a '''right coset of H''' in ''G''.

@@ Line 1: / Line 1: @@
 ==Definition==
-If G is a grup, H a of subgroup G, and g an element of G, then
+If G is a group, H a of [[subgroup]] G, and g an element of G, then
 :gH = { gh : h &isin; H } is a '''left coset of H''' in ''G''
 :Hg = { hg : h &isin; H } is a '''right coset of H''' in ''G''.

@@ Line 15: / Line 15: @@
 == Example ==
 The coset decomposition of the [[twin lattice]] point group with respect to the point group of the individual gives the different possible [[twin law]]s. Each element in a coset is a possible [[twin operation]].
 [[Category:Fundamental crystallography]]