Group
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 |
|---|---|---|
| 0 | - | - |
| 1 | E | - |
| 2 | C2 | - |
| 3 | C3 | - |
| 4 | C4 ; C2×C2 | - |
| 5 | C5 | - |
| 6 | C6 | D6 |
| 7 | C7 | - |
| 8 | C8 ; C4×C2 ; C2×C2×C2 | D8 ; Q |
| 9 | C9 ; C3×C3 | - |
| 10 | C10 | D10 |
| 11 | C11 | - |
See also: Alternating Group, An; Cyclic Group, Cn; Dihedral Group, D2n; Symmetric Group, Sn.