Number of Homomorphisms from K4 to Z8

Introduction
To find the number of homomorphisms from the complete graph K4 to the cyclic group Z8, we need to understand the properties of both structures and their relationship.

Definition of Homomorphism
A homomorphism is a function between two algebraic structures that preserves the operations of the structures. In the case of groups, a homomorphism is a function that preserves the group operation.

Properties of K4
The complete graph K4 has 4 vertices and 6 edges. Since K4 is a complete graph, every vertex is connected to every other vertex. Therefore, each vertex in K4 is adjacent to three other vertices.

Properties of Z8
The cyclic group Z8 has 8 elements: {0, 1, 2, 3, 4, 5, 6, 7}. It is closed under addition modulo 8, meaning that if we add two elements and take the remainder when divided by 8, the result is still an element of Z8.

Homomorphisms from K4 to Z8
To determine the number of homomorphisms from K4 to Z8, we need to consider the properties of both structures.

- A homomorphism must preserve the group operation, which is addition in Z8. Therefore, the image of each vertex in K4 must be an element of Z8.
- Since K4 has 4 vertices, there are 4 choices for the image of the first vertex, 4 choices for the image of the second vertex, and so on. This gives us a total of 4*4*4*4 = 256 possible homomorphisms.

Conclusion
In conclusion, there are 256 homomorphisms from the complete graph K4 to the cyclic group Z8. Each vertex in K4 can be mapped to any element in Z8, resulting in a total of 4*4*4*4 = 256 possible homomorphisms.