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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 ?

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  • $\begingroup$ The answer seems to be correct. Be prepared to give some more details though :-) $\endgroup$ – Jyrki Lahtonen Oct 19 '13 at 15:45
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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.

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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

$$x^2yx^{-1}y^{-1}x^3y^3=x^6y. $$

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