# Determine if morphism, find kernel and image

• polarbears
In summary, a morphism in mathematics is a structure-preserving mapping between two mathematical objects. To determine if a mapping is a morphism, it must preserve the structure of the objects being mapped. The kernel of a morphism is the set of elements in the domain that are mapped to the identity element in the codomain. To find the kernel, one must solve for the elements that satisfy the equation f(x) = e. The image of a morphism is the set of elements in the codomain that are mapped from the domain.
polarbears

## Homework Statement

Determine if the following is a group morphism. Find the kernel and the image if so.
$$f:C_{2} \times C_{3} \rightarrow S_{3}$$ where $$f(h^{r},k^{s})=(1,2)^{r} \circ (123)^{s}$$

## The Attempt at a Solution

I'm stuck on the morphism part. So I know I need to show that $$f(h^{r+x},k^{s+y})=(1,2)^{r+x} \circ (123)^{s+y} = (12)^{r} \circ (123)^{s} \circ (12)^{x} \circ (123)^{y}$$
but I'm not sure how to do that.
Also could someone check my set up?

Been awhile and no response

## 1. What is a morphism in mathematics?

A morphism is a term used in mathematics to describe a structure-preserving mapping between two mathematical objects. It can be thought of as a generalization of functions, where the two objects being mapped can have different structures.

## 2. How do you determine if a mapping is a morphism?

To determine if a mapping is a morphism, you need to check if it preserves the structure of the two objects being mapped. This means that the mapping should maintain the properties and relationships of the objects, such as operations and composition.

## 3. What is the kernel of a morphism?

The kernel of a morphism is the set of elements in the domain that are mapped to the identity element in the codomain. In other words, it is the pre-image of the identity element under the morphism.

## 4. How do you find the kernel of a morphism?

To find the kernel of a morphism, you need to determine which elements in the domain are mapped to the identity element in the codomain. This can be done by solving for the elements that satisfy the equation f(x) = e, where f is the morphism and e is the identity element in the codomain.

## 5. What is the image of a morphism?

The image of a morphism is the set of elements in the codomain that are mapped from the domain. In other words, it is the range of the morphism and contains all the elements that are mapped to from the domain.

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