Difference between revisions of "Group"
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| − | <font color="blue">Groupe</font> (''Fr'') | + | <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 | + | A set ''G'' equipped with a [[binary operation]] *: ''G × 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). | # 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> | + | # 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>. |
| − | Often, the symbol for the binary operation is omitted | + | 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]]. | + | 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'' | + | 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]]s are finite groups. | ||
==See also== | ==See also== | ||
| − | * | + | *Chapter 1.1 of ''International Tables for Crystallography, Volume A'', 6th edition |
[[Category:Fundamental crystallography]] | [[Category:Fundamental crystallography]] | ||
Revision as of 12:20, 15 May 2017
Groupe (Fr). Gruppe (Ge). Grupo (Sp). Gruppo (It). 群 (Ja).
A set G equipped with a binary operation *: G × 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 groups and their translation subgroups, whereas point groups are finite groups.
See also
- Chapter 1.1 of International Tables for Crystallography, Volume A, 6th edition