A set G, together with a biary operation •, which satisfy:
- Closure: x • y is in G, for all x,y in G.
- Identity: There exists e in G, such that x • e = e • x = x, for all x in G.
- Inverses: For all x in G, there exists an x' in G, such that x • x' = x' • x = e.
- Associativity: x • (y • z) = (x • y) • z, for all x,y,z in G.
Technically, G is the set and the group itself should be denoted 〈G,•〉. However, for brevity or where the intended operation is obvious, it is common to refer simply to the group G.
A group is said to be abelian if all its elements commute; in other words, if a • b = b • a for all a,b in G.
The order of a group, G, is the number of elements in the set G. It is usually witten as |G|, using standard set notation. When the order of the group is small, there are only a few possibilities for the structure which the group can have. Groups which have exactly the same structure are said to be isomorphic to one another.
|Order||Abelian Groups||Non-Abelian Groups|
|4||C4 ; C2×C2||-|
|8||C8 ; C4×C2 ; C2×C2×C2||D8 ; Q|
|9||C9 ; C3×C3||-|