A group is a set of elements paired with a binary operation. Four conditions called the group axioms must
Closure: If A and B are both members of the set then the result of A op B is also a member.
Associativity: Invocation/execution order doesn't matter - ((A op B) op C) == (A op (B op C))
The identity property: There is an identity element in the set, I, so that I op A == A op I == A
The inverse property: For each element in the set there must be an inverse element (opposite or
reciprocal) so that A-1 op A == A op A-1 == I
Note that commutativity is not a requirement - A op B doesn't always have to be equal to B op A. If the
operation is commutative then the group is called an abelian group or simply a commutative group.