# Understand the free group universal property applied to $D_n$

For $$n ≥ 3$$ and $$D_n$$ the dihedral group of order $$2n$$ with présentation $$\langle r, s : r^n = s^2 = srsr = 1\rangle$$

prove that for all $$(a, b) \in (\Bbb Z/n\Bbb Z)^2$$, there exists a morphism $$f$$ vérifying $$f(r) = r^a , f (s) = r^b s$$.

in the "hint" solution I have, it mentions the following:

The universal properties of free groups and the quotient show that $$f$$ is well defined if $$r^a$$ and $$r^b$$ s verify the relations of $$r$$ and $$s$$.

I don't understand, what is exactly meant by the free group? is it $$\Bbb Z$$ or $$D_{2n}$$. I know what is a free group but I can't really connect the dots.

Thanks for any help.

• The morphism $f$ is from where to where? – giannispapav Apr 15 at 17:15
• The free group is the free group on the letters $r$ and $s$. $D_n$ is the quotient of this free group by the relations $r^n, s^2,$ and $srsr$. – jgon Apr 15 at 17:23
• It is not stated clearly because I've written the question litteraly from my exam and I guess $f$ is $D_n \to D_n$ where $n$ is the Dihedral group of order $2n$ – PerelMan Apr 15 at 17:28