### Instruction: Statement 1 | The set of all integers under addition is an abelian group. Statement 2 | If G is a group and H is a subgroup of G, then the order of H divides the order of G.  Answer Choices: (A) True, True (B) False, False (C) True, False (D) False, True.### Response: Statement 1: The set of all integers under addition is an abelian group. This statement is true. The set of all integers (Z, +) forms a group because it satisfies the four group axioms: 1. Closure: If a and b are integers, then a + b is also an integer. 2. Associativity: For all integers a, b, and c, (a + b) + c = a + (b + c). 3. Identity element: The integer 0 is the identity element because for any integer a, a + 0 = a. 4. Inverse: For any integer a, there is an integer -a such that a + -a = 0. The group is abelian because the operation of addition is commutative: for all integers a and b, a + b = b + a. Statement 2: If G is a group and H is a subgroup of G, then the order of H divides the order of G. This statement is false. It is known as Lagrange's theorem and is true if G is a finite group. However, if G is infinite, then the order of G is not defined, and the statement is false. For example, consider the group of integers under addition (which is infinite), and its subgroup of even integers. The order of the subgroup is also infinite, and infinite does not divide infinite in a meaningful way. Therefore, the first statement is true and the second statement is false, The answer is B.