Difference between revisions of "Group"
From Online Dictionary of Crystallography
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*Section .1 in the ''International Tables for Crystallography, Volume A''
Revision as of 15:46, 10 April 2017
Groupe (Fr); Gruppe (Ge); Grupo (Sp); Gruppo (It); 群 (Ja).
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 groups and their translation subgroups, whereas point groups are finite groups.
- Section 1.1 in the International Tables for Crystallography, Volume A, 6th edition