Since $\mathbb{Z}_{2n}$ has elements of order $2n$, while $D_{2n}$, a dihedral group of order $2n$, only has elements of maximum order $n$, there is no isomorphism between the two groups. But I got confused.
If two groups are of the same order, then there must exist a bijection between the groups. And one can define a homomorphism, which does not necessarily preserve orders of elements between the groups. Then why does this homomorphism have to be non-injective?
Let $g$ be an element of $\mathbb{Z}_{2n}$ of order $2n$. So $g^0$, $g^1$, $g^2$, ..., $q^{2n-1}$ are distinct. But if there is a homomorphism $\phi$ from $\mathbb{Z}_{2n}$ to some subgroup of $D_{2n}$, then, since $\phi(g)$ has order less than $2n$, $\phi(g^0)$, $\phi(g)$, $\phi(g^2)=[\phi(g)]^2$, ..., $\phi(g^{2n-1})=[\phi(g)]^{2n-1}$ cannot be distinct. Hence $\phi$ is not injective.