Question

4. Recall that the symmetry group of rotations in 3-dimensions of the regular pentagon has 10 elements, which we denoted in class by 4: Consider this symmetry group acting on the set of coloured 5-bead necklaces in which each of the five beads is coloured white, grey or black. The figure below shows an example of such a necklace, which we refer to as A (a) Determine|orb(A)| (1 mark) (b) Determine |Stab(A)|. (1 mark)

Answer 1

(a) clearly 4 4 times rotation gives co=720 distinct such figures and Ro t h fixes it. So till now there are five distinct figures coming from Ro, R72 ... Ra88 . Now any reflection will give any one of above five figures because we have covered all circles which black balls can aquire a hus lorb (A) = 5. (b). Observe that For = SR is reflection of the below figure to wh which is the original figure thus Istab(A) | Ro & F4 € Stab (A).