To find the number of distinct group homomorphisms from (Z, +) onto (Z, +), we need to determine the possible mappings that preserve the group structure between the two groups.


A group homomorphism is a function between two groups that preserves the group operation. In this case, we are looking for a homomorphism from the additive group of integers (Z, +) to itself.


In order for a function to be a group homomorphism, it must satisfy the following properties:
1. Preserves the group operation: f(a + b) = f(a) + f(b) for all a, b in the domain group.
2. Preserves the identity element: f(0) = 0, where 0 is the identity element of the domain group.
3. Preserves inverses: f(-a) = -f(a) for all a in the domain group.


To find the group homomorphisms from (Z, +) onto (Z, +), we need to consider how the group operation is preserved.

Since the group operation in both (Z, +) and (Z, +) is addition, any homomorphism must satisfy the property f(a + b) = f(a) + f(b).

Let's consider the possible mappings for a group homomorphism:

1. Identity Mapping: f(x) = x for all x in Z. This mapping preserves the group operation as f(a + b) = a + b = f(a) + f(b). Therefore, it is a valid group homomorphism.
2. Constant Mapping: f(x) = c for all x in Z, where c is a constant integer. This mapping also preserves the group operation as f(a + b) = c = f(a) + f(b). Therefore, it is also a valid group homomorphism.


There are two distinct group homomorphisms from (Z, +) onto (Z, +): the identity mapping and the constant mapping. These mappings preserve the group operation and satisfy the properties of a group homomorphism.