
Geometric symmetries
This question tests your ability to describe symmetries geometrically and to represent them as permutations in cycle form. It also tests your understanding of conjugacy classes and their relationship to normal subgroups.
The figure for this question is prism with three identical rectangular faces and an equilateral triangle at the top and base. The locations of the faces of the prism (numbered 1, 4 at the top and base and each side 2, 3 and 5, 5 being the back face) have been numbered so that we may represent the group G of all symmetries of the prism as permutations of the set {1,2,3,4,5}.
a) Describe geometrically the symmetries of the prism represented in cycle form by (14)(23) and (25).
b) Write down all the symmetries of the prism in cycle form as permutations of {1,2,3,4,5}, and describe each symmetry geometrically.
c) Write down the conjugacy classes of G.
d) Determine a subgroup of G of order 2, a subgroup of order 3, and a subgroup of order 4. In each case, state whether or not your choice of subgroup is normal, justifying your answer.