Show that there exist an injective homomorphism of dihedral group $D_n$ into $G$.

Question

Let:
$$n\gt2 \; \text{and the group} \; (G,⋅)$$
Consider that there existe:

$a,b∈G$ such that $a^n=b^2=1_G$ and $b⋅a=a^{−1}⋅b$

and $n$ is the smallest $n≥1$, such that $a^n=1_G$.

Show that there exist an injective homomorphism of dihedral group $D_n$ into $G$.

I know that the dihedral group $D_n$ has the same properties:

$a,b∈G$ such that $a^n=b^2=1_G$ and $b⋅a=a^{−1}⋅b$ as the group $(G,⋅)$.

How to find an injective homomorphism,
that would prove the existence of a monomorphism of dihedral group $D_n$ into $G$?

Does someone could help me?

Answer 1

Let: $$D_n=\langle x,y | x^n, y^2, (xy)^2 \rangle$$

We can define a homomorphism by giving the action on its generators,
so consider the homomorphism:
$$\phi: D_n \rightarrow G \; \text{given by} \; x \mapsto a, \; y \mapsto b$$

What can you say about $\phi$?