Let $D_n$ denote the dihedral group of order 2n. Let G be a finite group. Prove that if there is a nontrivial homomorphism from $D_n$ to G then the order of G is even.
I am thinking to use $D_n/ker(\phi)\simeq G$, but I am confuse to use this theorem because question is asking about homomorphism, not about isomorphism.
Anyone please suggest me some direction to think this question?
If $G$ were odd, then we get that any even order element of $D_n$ must be mapped to the identity, else its image would generate an order $2$ subgroup of $G$, contradicting Lagranges theorem.
So now we just need to exhibit generators of $D_n$ of order $2$, since the the kernel must contain these, giving that the kernel is all of $D_n$.
I'll leave finding them to you, since it depends on your description of $D_n$, but such generators of order $2$ do exist.