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An exercise of our first problem set in geometry is to find all homomorphisms $\varphi: C_4 \to V$ or $\psi: V \to C_4$, where $C_4$ is the cyclic group and $V$ the Kleinian group. I have no idea how to start with this exercise. Can anyone give me a hint?

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up vote 2 down vote accepted

If $g$ is a generator of $C_4$ then $\phi$ is completely given by specifying $\phi(g)\in V$; you must make sure that $\phi(g)^4=1$, but that is true for all elements of $V$. Hence you find four $\phi\colon C_4\to V$.

On the other hand, $\psi$ is given by specifying the values of the two generators. But thes values must be of order dividing 2, not arbitrary elements of $C_4$. There are two such elements, $1$ and $g^2$, hence you have $2\cdot 2$ homomorphisms $\psi\colon V\to C_4$.

But I assume you are expected to write them all down explicitly (i.e. give the complete list of values for each of th efour plus four maps). At least the above thiughts should help you how to construct the different homomoprphisms.

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