Groups of Transformations
Definition of Group and Subgroup
Group: A group is a set with a binary operation (*; e.g., composition) that satisfies the following four properties:
- Closure: A set is closed under a binary operation if, when any two elements of the set are combined under that binary operation, the result is another element of the set.
- Identity: The identity element in a set is an element, e, of that set such that, for all elements a in the set, e*a=a*e=a.
- Inverses: For any element a in the set, there exists an inverse element, a-1, such that a*a-1 = a-1 *a= e.
- Associativity: If a, b, and c are elements of the set, then (a*b)*c=a*(b*c).