1. ## Cube. Coloring. Isomorphism.

Let $G$ be a group of rigid motions of cube.

a) Show that $G = S_4$

b) Show that the alternating subgroup $A_4 \le S_4$ is isomorphic to the group
of rigid motions of regular
tetrahedron.

c)Find cycle index for both $S_4$ and $A_4$.

d) Determine pattern inventory of $m$ coloring of a set
$X = \{d_1,d_2,d_3,d_4\}$, where $d_i$ is the $i$ -th diagonal of
the cube, $I(c_1,c_2,...,c_m)$ (representing colors $1$ to $m$) for
both $S_4$ and $A_4$.

2. ## Re: Cube. Coloring. Isomorphism.

What do you need help on? If it's the whole question, I suggest you do some revision BEFORE attempting this question. Otherwise, I can give some hints.

a) Consider how $G$ acts on - either the set of 6 faces, the set of 12 edges, the set of 8 vertices of a cube. I recommend using the last one. Regardless of your consideration, use the orbit-stabliser theorem to show that $|G|=24$.

Now note that vertices which are opposite each other remain opposite of each other under any rigid motion. Thus $G$ acts on the set of pairs of opposite vertices, of which there are 4. This induces a homomorphism $G\to S_4$, and you just need to show that the map is injective (or surjective) since the orders of $G$ and $S_4$ agree.

b) Same idea as before. This time, consider the set of the triangular faces of the tetrahedron (this has cardinality 4). Orbit-stabilizer argument shows that the order of the group is 12. Since G acts on the 4 faces, you have a homomorphism $G\to S_4$. Now show that the kernel of this map has order 2.

I suggest you investigate these two parts before moving on to c) and d). I believe you would not need help with the latter two if you understood parts a and b (unless of course, you don't know what cycle indices are. In that case, please revise before attempting this question).