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I am having lots of trouble doing this problem because I have particularly poor visualization skills. (Or maybe haven't developed them well yet). I would appreciate any help on this math problem.

Here is the question:

Suppose a cube is oriented before you so that from your point of view there is a top face, a bottom face, a front face, a back face, a left face, and a right face. In the group $O$ of rotational symmetries of the cube, let $x$ rotate the top face counterclockwise $90$ degrees, and let $y$ rotate the front face counterclockwise $90$ degrees.

(a) Find $g \in O$ such that $gxg^{-1}=y$ and $g$ has order $4$. (b) Find $h \in O$ such that $hxh^{-1}=y$ and $h$ has order $3$. (c) Find $k \in O$ such that $kxk^{-1}=x$ and $k$ has order $1$.

Thanks in advance.

arabhi manachra
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    Hint: Order 4 suggests you should look for rotations around faces (axes orthogonal to two opposite faces), order three suggests sapcial diagonal (axis passing through diametrically opposite vertices), order 1 is a very strong hint. - I suggest you really play for some hands-on experience with a cube (a die or Rubik's or even a package/box that is not fully cubic) just to get a feeling for his stuff such as the effects of conjugation – Hagen von Eitzen Nov 03 '13 at 00:28

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