Cube. Coloring. Isomorphism.
Let be a group of rigid motions of cube.
a) Show that
b) Show that the alternating subgroup is isomorphic to the group
of rigid motions of regular
c)Find cycle index for both and .
d) Determine pattern inventory of coloring of a set
, where is the -th diagonal of
the cube, (representing colors to ) for
both and .
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 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 .
Now note that vertices which are opposite each other remain opposite of each other under any rigid motion. Thus acts on the set of pairs of opposite vertices, of which there are 4. This induces a homomorphism , and you just need to show that the map is injective (or surjective) since the orders of and 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 . 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).