determine the number of homomorphisms from $D_5$ to $\mathbb{R^*} $ and from $D_5$ to $S_4$ I'm trying to do these three exercises for my math study:
a) Determine the number of homomorphisms from $D_5$ to $\mathbb{R^*}$
b) Determine the number of homomorphisms from $D_5$ to $S_4$
c) give an injective homomorphism form $D_5$ to $S_5$
I think I solved a):
The amount of homomorphisms is 1. The group $D_5$ is generated by a rotations and reflections. every reflection has order 2, and the rotations have order 1(the rotation about nothing) and order 5(the 4 rotations left). so we have to sent the reflection to -1 or 1, because the order of $f(x)$ has to divide the order of $x$ and the rotation about 0 degrees has only 1 as option to send it to. Combining that gives us only 1 homomorphism.
For b), I think the answer is 6! = 720, because every reflection in $D_5$ has to go to a 2-cycle in $S_4$, because the 2-cycles generate $S_4$. There are six of them, so the number of different homomorphisms are 6!.
Can you tell me if this is correct, and explain c) to me?
Thanks in advance!

For part c) you can take the generators $r=(12345)$ and $s=(25)(34)$ of $D_5$ and map them via the identity to $S_5$. This yields an injective homomorphism $\phi\colon D_5\rightarrow S_5$. (Note that $srs=s^{-1}$).
