Find all the homomorphisms from Z12 to Z6?

[kara1424]

How do you find all the homomorphisms from Z12 to Z6? and classify them by their kernals?

 

[matt grime]

Z12 is, I presume is the cyclic group with twelve elements. It is generated by a single element, 1. Where can 1 be sent to in Z6?

 

[M98Ranger]

To find all the homomorphisms from Z12 to Z6, we need to understand the properties of homomorphisms and how they relate to the group structures of Z12 and Z6.

A homomorphism is a function that preserves the group structure, which means that it maps the group operation of one group to the other. In this case, we are looking for homomorphisms from Z12 to Z6, so our function should map the addition operation in Z12 to the addition operation in Z6.

We can represent Z12 and Z6 using their cyclic group structures, where Z12 is generated by {1} and Z6 is generated by {1}. This means that any element in Z12 can be written as a multiple of 1, and any element in Z6 can be written as a multiple of 1.

Now, let's consider the possible homomorphisms from Z12 to Z6. Since our function should map the addition operation, we can start by looking at how the generator 1 in Z12 is mapped to Z6. Since Z6 is generated by 1, there are only 6 possible images of 1 in Z6: {0, 1, 2, 3, 4, 5}.

This means that there are 6 possible homomorphisms, each determined by where we map the generator 1 in Z12. Let's label these homomorphisms as f0, f1, f2, f3, f4, and f5, where fi maps 1 to i in Z6.

Now, we need to consider the kernal of each of these homomorphisms. The kernal of a homomorphism is the set of elements in the domain that are mapped to the identity element in the codomain. In this case, the identity element in Z6 is 0.

We can see that the kernal of f0 is {0, 6}, since these are the elements in Z12 that map to 0 in Z6. Similarly, the kernal of f1 is {0, 2, 4, 6, 8, 10}, the kernal of f2 is {0, 4, 8}, the kernal of f3 is {0, 6}, the kernal of f4 is {0, 2, 4, 6, 8