# Geometric symmetries

• Mar 26th 2013, 08:39 AM
raggie29
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.
• Mar 26th 2013, 11:13 AM
HallsofIvy
Re: Geometric symmetries
1) You should not just post a problem without any indication of what you have tried yourself.
2) You should not do it twice!
• Mar 26th 2013, 11:18 AM
raggie29
Re: Geometric symmetries
Sorry i was not exactly sure what category to post the question in.
• Mar 26th 2013, 11:09 PM
hollywood
Re: Geometric symmetries
I think the question belongs in Advanced Algebra.

I will make one rather obvious comment: the faces are of two types - triangles and rectangles. Triangles have to map to triangles and rectangles have to map to rectangles, so you have only permutations of the set {1,4} combined with permutations of the set {2,3,5}, but perhaps not even all of those.

So what does the permutation (14)(23) look like for the prism? What about (25)?

- Hollywood
• Mar 26th 2013, 11:11 PM
hollywood
Re: Geometric symmetries
Quote:

Originally Posted by HallsofIvy
2) You should not do it twice!

And certainly not 3 times!!