The set of integers is an abelian group under?

The set of integers is an abelian group under?

Explanation

The set of integers, denoted by Z, forms an abelian group under the operation of addition (+). This means that the following properties hold

  • Closure: For any integers a, b, their sum a + b is also an integer.
  • Associativity: (a + b) + c = a + (b + c) for any integers a, b, c.
  • Commutativity: a + b = b + a for any integers a, b.
  • Identity: There exists an identity element, which is 0, such that a + 0 = a for any integer a.
  • Inverse: For each integer a, there exists an inverse element, which is -a, such that a + (-a) = 0.

The set of integers does not form a group under multiplication, because the inverse property does not hold (except for 1 and -1). Division is not even a binary operation on the set of integers, because it is not defined for all pairs of integers (e.g., 1/2 is not an integer).

Therefore, the correct answer is Addition.