Week 5 HW, Chapter 6, Problem 7, MA453, Spring 2008, Prof. Walther
Problem Statement
Prove that S4 is not isomorphic to D12.
Discussion
D12 has elements of order 12 whereas S4 does not and therefore they cannot be isomporphic. --Aifrank 17:05, 9 February 2009 (UTC)
What do the elements of D12 and S4 look like? I found some pictures of what D12 and S4 can look like, but I am really stuck on what the elements are. Are they the rotations and reflections? --Josie
Yes they are the rotations and reflections. There are 12 rotations (30 degrees each - 360/12) as well as reflections. So you can see that D12 has elements of order 12 from those rotations and reflections. Then in S4 you either have a 4-cycle, or a 3-cycle, or 2 2-cycles, or 1 2-cycle. The orders of these are 4, 3, 2, 2. So none of them are order 12.
--Nswitzer 16:37, 10 February 2009 (UTC)
Thank you very much for the help! --Josie
Can you also prove this one by showing the size of S4 isn't equal to the size of D12? -Paul
If by size you mean the order, then $ \scriptstyle\mid S_4\mid $ is actually equal to $ \scriptstyle\mid D_{12}\mid $, their orders being 24.
- --Nick Rupley 02:57, 12 February 2009 (UTC)
Yeah, would have been nice if D12 and S4 as groups had the same order...but they don't so you have to do it that way
