Question on the definition of the Dihedral groups

Question

I have seen the definition of the dihedral group of order $2n$ in several places as $$ D_n = \langle x,y \mid x^n = y^2 = e, yxy^{-1} = x^{-1}\rangle. $$

My question is why there is an "inverse" on the second $y$. Since $y^2 = e$ isn't $y = y^{-1}$? And if this is the case couldn't one just replace $yxy^{-1} = x^{-1}$ with $yxy= x^{-1}$.

Am I missing something?

Answer 1

No, you are right, it does not make a difference. I think it's there to highlight that we are taking a conjugate, which could be less obvious with $yxy$.

Answer 2

Both are true, but what this definition wants to highlight is that $(xy)^{-1}=y^{-1}x^{-1} ≠ x^{-1}y^{-1}$ (It doesn't necessarily commute).