Group - Revision history

MassimoNespolo: not necessarily true for molecular point groups (ex. cylindrical molecules like acetylene)

2018-11-30T15:46:34Z

not necessarily true for molecular point groups (ex. cylindrical molecules like acetylene)

BrianMcMahon: Tidied translations.

2017-11-13T14:43:57Z

Tidied translations.

MassimoNespolo: languages

2017-10-13T14:01:25Z

languages

BrianMcMahon: Style edits to align with printed edition

2017-05-15T12:20:52Z

Style edits to align with printed edition

MassimoNespolo: /* See also */ 6th edition of ITA

2017-04-10T15:46:01Z

‎See also: 6th edition of ITA

MassimoNespolo: Ja

2009-04-02T14:01:39Z

BerndSouvignier: initial definition

2009-04-01T14:43:21Z

initial definition

New page

Groupe (''Fr''); Gruppe (''Ge''); Grupo (''Sp''); Gruppo (''It'').

A set ''G'' equipped with a [[binary operation]] *: ''G x G'' → ''G'', assigning to a pair ''(g,h)'' the product ''g*h'' is called a '''group''' if:
# The operation is ''associative'', i.e. ''(a*b)*c = a*(b*c).
# ''G'' contains an ''identity element'' (''neutral element'') ''e'': ''g*e = e*g = g'' for all ''g'' in ''G''
# Every ''g'' in ''G'' has an ''inverse element'' ''h'' for which ''g*h = h*g = e''. The inverse element of ''g'' is written as ''g -1.

Often, the symbol for the binary operation is omitted, the product of the elements ''g'' and ''h'' is then denoted by the concatenation ''gh''.

The binary operation need not be commutative, i.e. in general one will have ''g*h ≠ h*g''. In the case that ''g*h = h*g'' holds for all ''g,h'' in ''G'', the group is an [[Abelian group]].

A group ''G'' may have a finite or infinite number of elements. In the first case, the number of elements of ''G'' is the '''order''' of ''G'', in the latter case, ''G'' is called an '''infinite group'''.
Examples of infinite groups are [[space group]]s and their translation subgroups, whereas [[point group]]s are finite groups.

==See also==

*Section 8.1.3 in the ''International Tables for Crystallography, Volume A''

[[Category:Fundamental crystallography]]

@@ Line 12: / Line 12: @@
 A group ''G'' may have a finite or infinite number of elements. In the first case, the number of elements of ''G'' is the '''order''' of ''G''. In the latter case, ''G'' is called an '''infinite group'''.
-Examples of infinite groups are [[space group]]s and their translation subgroups, whereas [[point group]]s are finite groups.
+Examples of infinite groups are [[space group]]s and their translation subgroups, whereas [[point group|crystallographic point groups]] are finite groups.
 ==See also==

← Older revision		Revision as of 14:43, 13 November 2017
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−	<font color="orange">زمرة</font> (''Ar''); <font color="blue">Groupe</font> (''Fr''); <font color="red">Gruppe</font> (''Ge''); <font color="black">Gruppo</font> (''It''); <font color="purple">群</font> (''Ja''); <font color="brown">Группа</font> (''Ru'').<font color="green">Grupo</font> (''Sp'').	+	<font color="orange">زمرة</font> (''Ar''). <font color="blue">Groupe</font> (''Fr''). <font color="red">Gruppe</font> (''Ge''). <font color="black">Gruppo</font> (''It''). <font color="purple">群</font> (''Ja''). <font color="brown">Группа</font> (''Ru''). <font color="green">Grupo</font> (''Sp'').

← Older revision		Revision as of 14:01, 13 October 2017
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−	<font color="blue">Groupe</font> (''Fr''). <font color="red">Gruppe</font> (''Ge''). <font color="~~green~~">~~Grupo~~</font> (''Sp''). <font color="~~black~~">~~Gruppo~~</font> (''It''). <font color="~~purple~~">群</font> (''Ja'').	+	<font color="orange">زمرة</font> (''Ar''); <font color="blue">Groupe</font> (''Fr''); <font color="red">Gruppe</font> (''Ge''); <font color="black">Gruppo</font> (''It''); <font color="purple">群</font> (''Ja''); <font color="brown">Группа</font> (''Ru'').<font color="green">Grupo</font> (''Sp'').

@@ Line 1: / Line 1: @@
-<font color="blue">Groupe</font> (''Fr''); <font color="red">Gruppe</font> (''Ge''); <font color="green">Grupo</font> (''Sp''); <font color="black">Gruppo</font> (''It''); <font color="purple">群</font> (''Ja'').
+<font color="blue">Groupe</font> (''Fr''). <font color="red">Gruppe</font> (''Ge''). <font color="green">Grupo</font> (''Sp''). <font color="black">Gruppo</font> (''It''). <font color="purple">群</font> (''Ja'').
-A set ''G'' equipped with a [[binary operation]] *: ''G x G'' &rarr; ''G'', assigning to a pair ''(g,h)'' the product ''g*h'' is called a '''group''' if:
+A set ''G'' equipped with a [[binary operation]] *: ''G &times; G'' &rarr; ''G'', assigning to a pair ''(g,h)'' the product ''g*h'' is called a '''group''' if:
 # The operation is ''associative'', i.e. ''(a*b)*c = a*(b*c).
-# ''G'' contains an ''identity element'' (''neutral element'') ''e'': ''g*e = e*g = g'' for all ''g'' in ''G''
+# ''G'' contains an ''identity element'' (''neutral element'') ''e'': ''g*e = e*g = g'' for all ''g'' in ''G''.
-# Every ''g'' in ''G'' has an ''inverse element'' ''h'' for which ''g*h = h*g = e''. The inverse element of ''g'' is written as ''g<sup> -1</sup>.
+# Every ''g'' in ''G'' has an ''inverse element'' ''h'' for which ''g*h = h*g = e''. The inverse element of ''g'' is written as ''g<sup>&minus;1</sup>.
-Often, the symbol for the binary operation is omitted, the product of the elements ''g'' and ''h'' is then denoted by the concatenation ''gh''.
+Often, the symbol for the binary operation is omitted. The product of the elements ''g'' and ''h'' is then denoted by the concatenation ''gh''.
-The binary operation need not be commutative, i.e. in general one will have ''g*h &ne; h*g''. In the case that ''g*h = h*g'' holds for all ''g,h'' in ''G'', the group is an [[Abelian group]].
+The binary operation need not be commutative, ''i.e.'' in general one will have ''g*h &ne; h*g''. In the case that ''g*h = h*g'' holds for all ''g,h'' in ''G'', the group is an [[Abelian group]].
-A group ''G'' may have a finite or infinite number of elements. In the first case, the number of elements of ''G'' is the '''order''' of ''G'', in the latter case, ''G'' is called an '''infinite group'''.
+A group ''G'' may have a finite or infinite number of elements. In the first case, the number of elements of ''G'' is the '''order''' of ''G''. In the latter case, ''G'' is called an '''infinite group'''.
 Examples of infinite groups are [[space group]]s and their translation subgroups, whereas [[point group]]s are finite groups.
 ==See also==
-*Section 1.1 in the ''International Tables for Crystallography, Volume A'', 6<sup>th</sup> edition
+*Chapter 1.1 of ''International Tables for Crystallography, Volume A'', 6th edition
 [[Category:Fundamental crystallography]]

← Older revision		Revision as of 14:01, 2 April 2009
Line 1:		Line 1:
−	<font color="blue">Groupe</font> (''Fr''); <font color="red">Gruppe</font> (''Ge''); <font color="green">Grupo</font> (''Sp''); <font color="black">Gruppo</font> (''It'').	+	<font color="blue">Groupe</font> (''Fr''); <font color="red">Gruppe</font> (''Ge''); <font color="green">Grupo</font> (''Sp''); <font color="black">Gruppo</font> (''It''); <font color="purple">群</font> (''Ja'').