A set G, together with a biary operation •, which satisfy:

  1. Closure: xy is in G, for all x,y in G.
  2. Identity: There exists e in G, such that xe = ex = x, for all x in G.
  3. Inverses: For all x in G, there exists an x' in G, such that xx' = x' • x = e.
  4. Associativity: x • (yz) = (xy) • 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 ab = ba 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.