A set of elements that can be combined by a binary operation and must satisfy the following conditions:
- The set must be closed under the operation;
- The operation is associative;
- It must have an identity element;
- Every element of the set must have an inverse element.
The group is called a commutative group or Abelian group if the operation is commutative.
The integers form a group under the addition operation because:
- The set is closed under the addition operation because adding one integer to another integer always gives an integer (the property of closure);
- The addition operation is associative because a + (b + c) = (a + b) + c holds true for any integers a, b, and c;
- It has an identity element, zero. Adding zero to any integer leaves it unchanged;
- Every integer a has an inverse, which is (-a).