Finding Homomorphisms

Question

Let D be the dihedral group of order 6 and C be the cyclic group of order 6. Find all homomorphisms D $\longrightarrow$C, (you may assume standard facts about cyclic and dihedral groups but you must state them), any help is greatly appreciated thank you!

Answer 1

Let $D$ be generated by $a$ and $b$ with $a^3=b^2=1$ and $bab=a^2$. The kernel of any morphism is normal in $D$. The only normal subgroups of $D$ are $D$ itself, the trivial subgroup and $\{1, a, a^2\}$. If the kernel is trivial then our morphism would be an isomorphism, which is impossible. If the kernel is $D$ then we have the zero map. If the kernel is $\{1, a, a^2\}$ then $b$ must be mapped to a non-zero element of order $2$. Hence $b\mapsto 3$ and $a\mapsto 0$.