Answer:

To find the number of distinct group homomorphisms from (Z, +) onto (Z, +), we need to consider the properties of group homomorphisms and the structures of the given groups.

Group Homomorphisms:
A group homomorphism is a function between two groups that preserves the group operations. More formally, let (G, *) and (H, •) be two groups. A function f: G -> H is a group homomorphism if it satisfies the following condition for all elements a, b in G:
f(a * b) = f(a) • f(b)

The Groups (Z, +) and (Z, +):
The group (Z, +) consists of the integers under addition. It is an abelian group, meaning that the group operation (addition) is commutative. Every integer can be represented in the form of a * 1, where a is an integer. Therefore, the function f: Z -> Z defined by f(a) = a is a group homomorphism.

Determining the Number of Group Homomorphisms:
To find the number of distinct group homomorphisms from (Z, +) onto (Z, +), we need to determine if there are any other possible group homomorphisms besides the identity function.

Identity Function:
The identity function f(a) = a is a group homomorphism from (Z, +) to (Z, +) as it preserves the group operation.

No Other Group Homomorphisms:
There are no other distinct group homomorphisms from (Z, +) onto (Z, +). This can be proven by considering the properties of group homomorphisms.
- For a function to be a group homomorphism, it must preserve the group operation. In this case, the group operation is addition.
- Any other function that satisfies f(a * b) = f(a) • f(b) will not be a group homomorphism.
- Since the identity function is already a group homomorphism, any other function that maps integers to integers in a different way will not preserve addition and thus will not be a group homomorphism.

Therefore, there is only one distinct group homomorphism from (Z, +) onto (Z, +), which is the identity function. Hence, the correct answer is option 'B' (2).