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Orders of elements for rotational symmetries of cube

kalish

Member
Oct 7, 2013
99
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.
 
Last edited:

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,794
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?
 

kalish

Member
Oct 7, 2013
99
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!