Can anyone help with the following problem:

Find all possible homomorphisms between the indicated groups:

$\displaystyle \phi $: $\displaystyle S_3 \rightarrow Z_6$

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Thoughts:

The only normal subgroups of $\displaystyle S_3$ are {e} and $\displaystyle A_3$

Thus (following the case for {e}) we can assume Ker $\displaystyle \phi $ = {e} for a homomorphism $\displaystyle \phi $ (by the First Isomorphism Theorem)

But how do we find the find the homomorphism - or homomorphisms?

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If I can see the situation for {e} then I am hoping a similar analysis will yield the homomorphism(s) for the case of $\displaystyle A_3 $

Be grateful for some help.

Peter