# Determine if morphism, find kernel and image

1. Nov 25, 2009

### polarbears

1. The problem statement, all variables and given/known data
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 $$f(h^{r},k^{s})=(1,2)^{r} \circ (123)^{s}$$

2. Relevant equations

3. 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?

2. Feb 15, 2010

### polarbears

Been awhile and no response

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