The number of group homomorphisms from the group Z4 to the group SE

Definition of a group homomorphism
A group homomorphism is a function between two groups that preserves the group structure. In other words, if we have two groups G and H, a function f: G -> H is a group homomorphism if it satisfies the following condition:
- f(a * b) = f(a) * f(b) for all elements a, b in G, where * denotes the group operation.

The group Z4
The group Z4 is the set of integers modulo 4, which can be represented as {0, 1, 2, 3}. The group operation is addition modulo 4, denoted as +. For example, 2 + 3 = 1 (mod 4).

The group SE
The group SE refers to the group of symmetries of an equilateral triangle. It consists of 6 elements: the identity element, 3 rotations (clockwise by 120 degrees, 240 degrees, and 0 degrees), and 2 reflections.

Counting the number of group homomorphisms
To count the number of group homomorphisms from Z4 to SE, we need to determine the images of the elements of Z4 under a homomorphism.

Images of the identity element
The identity element of Z4 is 0. Since the identity element of a group is always mapped to the identity element of the target group, the image of 0 under any homomorphism from Z4 to SE is the identity element of SE.

Images of non-zero elements
The non-zero elements of Z4 are {1, 2, 3}. Let's consider the image of 1 under a homomorphism.

- The order of 1 in Z4 is 4, which means that 1^4 = 1 (mod 4). Since the order of an element in the source group must divide the order of the element in the target group, the image of 1 must have an order of 1, 2, 3, or 4 in SE.
- The only elements in SE with an order of 1 are the identity element and the rotation by 0 degrees.
- The elements in SE with an order of 2 are the rotations by 120 degrees and 240 degrees.
- The elements in SE with an order of 3 are the rotations by 120 degrees and 240 degrees.
- The element in SE with an order of 4 is the identity element.

Conclusion
Considering all the possibilities for the image of 1, we see that there are 2 elements in SE with an order of 1, 2 elements with an order of 2, 2 elements with an order of 3, and 1 element with an order of 4. Therefore, there are a total of 2 * 2 * 2 * 1 = 8 group homomorphisms from Z4 to SE.