1.50: Group
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.