Number of group homomorphisms from Z4 to Z7

To find the number of group homomorphisms from the cyclic group Z4 to the cyclic group Z7, we need to understand the properties of group homomorphisms and their relationship with the orders of the groups involved.

Definition of a group homomorphism:
A group homomorphism is a function between two groups that preserves the group operation. In other words, if we have two groups G and H with the respective group operations *, then a function f: G -> H is a group homomorphism if for all elements a and b in G, f(a * b) = f(a) * f(b).

Properties of group homomorphisms:
- The identity element of the group G must be mapped to the identity element of the group H.
- The order of the element in G must divide the order of the element in H.

Order of a cyclic group:
The order of a cyclic group is equal to the number of elements in the group. For a cyclic group Zn, the order is n.

Solution:
- The cyclic group Z4 has 4 elements: {0, 1, 2, 3}.
- The cyclic group Z7 has 7 elements: {0, 1, 2, 3, 4, 5, 6}.

In order for a function to be a group homomorphism, the identity element of Z4 (0) must be mapped to the identity element of Z7 (0).

Mapping of 0:
Since the identity element of Z4 is 0, the mapping of 0 must be 0 in Z7.

Mapping of other elements:
Since the order of an element in Z4 must divide the order of the corresponding element in Z7, we need to find the possible mappings for the other elements.

- Mapping of 1: The order of 1 in Z4 is 4, and the order of the corresponding element in Z7 must divide 7. The only possible mapping is 1 -> 1.
- Mapping of 2: The order of 2 in Z4 is 2, and the order of the corresponding element in Z7 must divide 7. The possible mappings are 2 -> 1 or 2 -> 6.
- Mapping of 3: The order of 3 in Z4 is 4, and the order of the corresponding element in Z7 must divide 7. The only possible mapping is 3 -> 3.

Total number of group homomorphisms:
Considering all the possible mappings, we have:
- 0 -> 0
- 1 -> 1
- 2 -> 1 or 2 -> 6
- 3 -> 3

Therefore, there is only one possible group homomorphism from Z4 to Z7.

Conclusion:
The correct answer is option D) 1. There is only one group homomorphism from the cyclic group Z4 to the cyclic group Z7.