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Find the number of group homomorphisms between $C_6$ and $S_3$.

For all the group theory buffs this is probably a piece of cake, but how does one generally go about a question like this. Is there a general way to figure this out, or do you need to make use of case specific counting arguments, or group specific characteristics? Any tips would be very helpful!

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    Homomorphic images of a cyclic group is cyclic (why?). What are the possible homomorphic images of $C_6$? (You have four). Which of these images correspond to subgroups of $S_3$? This is the same as "does there exist an element of order $x$", as $C_x\leq G$ if and only if $G$ contains an element of order $x$. Then, how many such elements are there? For example, $S_3$ has one subgroup of order $3$, but this subgroup has two generators. This corresponds to two homomorphic images.2012-06-26

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Suppose $G$ is any finite cyclic group--generated by $g$, say--and $H$ is any other group. If $\varphi:G\to H$ is a homomorphism, then the order of $\varphi(g)$ must divide the order of $g$.

Now, given some appropriate $h\in H$ (in the finite $|G|$ case, such that the order of $h$ divides the order of $g$), we can define a function $\varphi_h:G\to H$ by $\varphi_h(g^k)=h^k$. This will be a homomorphism.

In fact, all such homomorphisms $\varphi:G\to H$ must have this form, for one can readily show by induction that $\varphi(g^k)=\varphi(g)^k$ for all integers $k$. Thus, the situation is completely described.

Note that in the particular case you've described, all $h\in H$ are appropriate, so fixing $g$ as one of the two possible generators of $C_6$, we may describe $6$ homomorphisms $C_6\to S_3$ as described above.

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Any such homomorphism is completely determined by what it does to a generator of $C_6$. Can you take it from there?

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    @BallzofFury: Any element of $S_3$ whose order is a divisor of $6$ (which, as it happens, is all of them); yes.2012-06-26