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

a) Show that $\displaystyle G = S_4$

b) Show that the alternating subgroup $\displaystyle A_4 \le S_4$ is isomorphic to the group

of rigid motions of regular

tetrahedron.

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

d) Determine pattern inventory of $\displaystyle m$ coloring of a set

$\displaystyle X = \{d_1,d_2,d_3,d_4\}$, where $\displaystyle d_i$ is the $\displaystyle i$ -th diagonal of

the cube, $\displaystyle I(c_1,c_2,...,c_m)$ (representing colors $\displaystyle 1$ to $\displaystyle m$) for

both $\displaystyle S_4$ and $\displaystyle A_4$.

Please, help!