Find two distinct group homomorphisms between $(U_{11}, *)$ and $(\mathbb{Z}_{10}, +)$, where $U_{11} =$ set of units in $\mathbb{Z}_{11}$.
My observations: We want to create a function $f: (U_{11}, *) \to (\mathbb{Z}_{10}, +)$ such that $f(n) = f(a*b) = f(a)+f(b)$, where $n = a*b$ ($a,b,n \in U_{11}$). With respect to $*$, $U_{11}$ is a cyclic group: $U_{11}= \langle2\rangle =\langle6\rangle=\langle7\rangle=\langle8\rangle$. Similarly, with respect to $+$, $\mathbb{Z}_{10}$ is cyclic: $\mathbb{Z}_{10}=\langle1\rangle=\langle3\rangle=\langle7\rangle=\langle9\rangle$.
I am not sure how to proceed after this. Any help would truly be appreciated.
A homomorphism is uniquely determined by where it sends the elements of some generating set. So take, for instance, the generator $2\in U_{11}$, and send it to different generators of $\Bbb Z_{10}$ to get different isomorphisms.