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For each of the following elements in $D_{14}$, write in canonical form:

  1. $x^6x^5yx^3xyyyx^{−2}$
  2. $(x^2yx^{−3})^2 $
  3. $(yx^{-7}x^2yyx^4 )^{−1}$

I've looked everywhere online and in my text book and have found nothing that helps with this. If anybody has any links to anything dealing with this or pointers please let me know!

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    Please use [MathJax](http://meta.math.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference). – 2017-02-11
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    What does "canonical form" mean for you? Something like $x^ay^b$? – 2017-02-11

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Note that $yxy = x^{-1}$, and more generally $yx^ny = (yxy)(yxy)(yxy)\cdots (yxy) = x^{-n}$. ($y$ has order two.) We can use these relations, along with the derived relations $yx^n = x^{-n}y$ (again, $y$ has order two) to shift all the $x$ to the left (or right, if you want, depending on your definition of "canonical expression") of some expression. For example:

The first one is equivalent to $x^{11} y x^4 y^3 x^{-2}$. Applying the relation $yx^4y = x^{-4}$ we get that this is equivalent to $x^{11}x^{-4}y^2x^{-2} = x^7 y^2 x^{-2}$. Applying the relation $yx^{-2} = x^2 y$ to this we get that this is equivalent to $x^7 yx^2 y$. Finally, applying $yx^2 y = x^{-2}$ we find that the original expression is just $x^7 x^{-2} = x^5$.