1. The set must be closed under the operation;
2. The operation is associative;
3. It must have an identity element;
4. 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:

1. The set is closed under the addition operation because adding one integer to another integer always gives an integer (the property of closure);
2. The addition operation is associative because a + (b + c) = (a + b) + c holds true for any integers a, b, and c;
3. It has an identity element, zero. Adding zero to any integer leaves it unchanged;
4. Every integer a has an inverse, which is (-a).