(i) State the orbit-stabilizer theorem for a group acting on a set .
(ii) Denote the group of all symmetries of the cube by . Using the orbit-stabilizer theorem, show that has 48 elements.
Does have any non-trivial normal subgroups?
Let denote the line between two diagonally opposite vertices of the cube, and let
be the subgroup of symmetries that preserve the line. Show that is isomorphic to the direct product , where is the symmetric group on 3 letters and is the cyclic group of order 2 .