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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?

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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 $\phi$?

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  • $\begingroup$ I haven't seen the generator concept yet. Is it possible to write with a somewhat simpler notation? $\endgroup$ – user230283 Sep 23 '15 at 18:22
  • $\begingroup$ Are you happy with my notation for $D_n$ and that every element can be written as a product of $x$ and $y$? $\endgroup$ – Matt B Sep 23 '15 at 18:30
  • $\begingroup$ No, I'm not happy with your notation for $D_n$ $\endgroup$ – user230283 Sep 23 '15 at 18:35
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    $\begingroup$ Ok how do you define $D_n$ then? $\endgroup$ – Matt B Sep 23 '15 at 18:41
  • $\begingroup$ Like this : $D_n = \{ \begin{array}{lc} \left(\begin{matrix} \cos2\pi k / n & -\sin2\pi k / n\\ \sin2\pi k / n& \cos2\pi k / n \end{matrix}\right) \\\end{array}, k \in \mathbb{Z} \} \cup \{\begin{array}{lc} \left(\begin{matrix} \cos2\pi k / n & \sin2\pi n \\ \sin2\pi n / 4& -\cos2\pi k/n \\ \end{matrix}\right) \\\end{array}, k \in \mathbb{Z} \}$, the set of rotations and reflections $\endgroup$ – user230283 Sep 23 '15 at 20:37

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