A group is a set, G, together with an operation • (called the group law of G) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a group, the set and operation, (G, •), must satisfy four requirements known as the group axioms
Closure
For all a, b in G, the result of the operation, a • b, is also in G.
Associativity
For all a, b and c in G, (a • b) • c = a • (b • c).
Identity element
There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique, and thus one speaks of theidentity element.
Inverse element
For each a in G, there exists an element b in G, commonly denoted a^{−1} (or −a, if the operation is denoted "+"), such that a • b = b • a = e, where e is the identity element.
The result of an operation may depend on the order of the operands. In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation
a • b = b •
Abelian Group
An abelian group is a set, A, together with an operation • that combines any two elements a and b to form another element denoted a • b. The symbol • is a general placeholder for a concretely given operation. To qualify as an abelian group, the set and operation, (A, •), must satisfy five requirements known as the abelian group axioms:
Closure
For all a, b in A, the result of the operation a • b is also in A.
Associativity
For all a, b and c in A, the equation (a • b) • c = a • (b • c) holds.
Identity element
There exists an element e in A, such that for all elements a in A, the equation e • a = a • e = aholds.
Inverse element
For each a in A, there exists an element b in A such that a • b = b • a = e, where e is the identity element.
Commutativity
For all a, b in A, a • b = b • a.
A group in which the group operation is not commutative is called a "non-abelian group" or "non-commutative group"