Group theory

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:

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.

Inverse element: For each a in G, there exists an element b in G such that a • b = b • a = e, where e is the identity element.

Groups for which the commutativity equation a • b = b • a always holds are called abelian groups.