Write the product $x^2yx^{-1}y^{-1}x^3y^3$ in the form $x^iy^j$ in the dihedral group $D_n$.
I used the fact that the dihedral group is generated by two elements $x$ and $y$ such that: $y^n=1$, $x^2=1$ and $xy=y^{-1}x$
and I found that $x^2yx^{-1}y^{-1}x^3y^3=y^5$
Is it correct ?
If your problem comes from Artin's algebra, then you have swapped the roles of $x$ and $y$. It should be
$$x^n=1, \quad y^2=1, \quad yx=x^{-1}y. $$
Thus I got
$$x^2yx^{-1}y^{-1}x^3y^3=x^6y. $$
Yes, I think, except that we can also write $$ x^2yx^{-1}y^{-1}x^3y^3 = 1yxy^{-1}xy^3 = yxxyy^3 = y1y^4 = y^5 = y^{(5 \mod n)}.$$ Here we have used the fact that $x^2=1$ and so $x^{-1} = x$, and $5 \mod n$ denotes the remainder when $5$ is divided by $n$ in accordance with the Euclidean division algorithm.
