MHB Orders of elements for rotational symmetries of cube

[kalish1]

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 $kxk^{-1}=x^{-1}$.

Thanks in advance.

 

[I like Serena]

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 $kxk^{-1}=x^{-1}$.

Thanks in advance.

Hi kalish!

Do you already have a list of the different rotational symmetries of the cube?
Obviously, g, h, and k have to be one of them.

Did you already make a drawing of a cube and mark x and y in it?

Typically g has to be of the form: turn the front face (that y acts on) somewhere where x can act on it (top face or bottom face), and afterward turn it back again.
This is called a conjugation and it is one of the main tricks to solve puzzles.
Which rotation of the cube will do that?
Can you make a drawing of the result?

If you have difficulty visualizing it, I propose you mark each face of the cube with a letter, say F, B, T, D, L, and R.
And then write a rotational symmetry as a combination of disjoint cycles.
For instance, rotating the top face counter clock wise by 90 degrees is: (F R B L).
That is, front goes to right, right goes to back, back goes to left, and left goes to front.

Are you familiar with applying cycles to each other?

 

[kalish1]

Hi *I like Serena*,
I am familiar with permutation cycles and disjoint cycles. I think your hints are well written. I will try to use them. Thanks!