Group Theory

Abstract

A group consists of a set (of symmetry operations, numbers, etc.) together with a rule by which any two elements of the set may be combined — which will be called, generically, multiplication — with the following four properties:

• 1 Closure: The result of combining any two elements — the product of any two elements — is another element in the set.

• 2 Group multiplication satisfies the associative law: a • (b • c) = (a • b) • c for all elements a, b and c of the group.

• 3 There exists a unit element, or identity, denoted E. such that E • a = a for any element of the group.

• 4 For every element a of the group, the group contains another element called the inverse, a-1, such that a • a-1 = E. Note that as E-E = E, the inverse of E is E itself.