Groups and special relativity

Fundamentals of group theory

Properties of groups

A group is a set of elements that have the following properties:

1. • Contains the identity transformation 1;

2. • If E and E′ are elements in the group, then there exists a group operation (generically called “multiplication”) that always produces an element of the group, E″ = E′ · E (closure);

3. • For every transformation element E, there exists in the group an inverse element E-1, where E-1 • E = 1;

4. • The group operation is associative, E′ · (E′ · E) = (E″ · E′) · E.

A particular type of group satisfies an additional property. For an abelian group, the group operation yields the same answer regardless of the order, E′ · E = E · E′. Often, a subset of the elements within a group satisfies all four group properties. This subset is referred to as a subgroup.

Generally, two different groups are isomorphic if there is a one-to-one relationships between the elements of the groups with regards to group operations. More generally, if a set of elements in one group are in direct relationship with one element in another, the groups are homomorphic.

A particular class of groups is quite useful for describing transformations in quantum physics. Invertible N × N matrices (i.e., matrices with non-vanishing determinants) form the set of linear groups.