A set of elements that can be combined by a binary operation and must satisfy the following conditions:

  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).

Related Term: commutative

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