Trying to understand D_n in two dimensions - sci.math

Message from discussion Trying to understand D_n in two dimensions

eratosthenes

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More options Nov 7 2009, 1:06 pm

Newsgroups: sci.math

From: eratosthenes <rehamkcir...@gmail.com>

Date: Sat, 7 Nov 2009 05:06:13 -0800 (PST)

Local: Sat, Nov 7 2009 1:06 pm

Subject: Trying to understand D_n in two dimensions

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In the dihedral group of order n where n is greater than or equal to 3
there are two general cases, n is odd or n is even

In both cases n mappings from D to itself exist as rotations of (2*pi)/
n

When n is odd there are also n mappings that exist as what I visualize
as "flips" (a rotation of pi radians about a vertex). In terms of
permutations I can write these for D_3 as (ABC) -> (ACB), (ABC) ->
(CBA) and (ABC) -> (BAC).

The situation for flips is similar for n even but the n/2 axes of
symmetry for the "flips" are from center to center of diametrically
opposed flats. Again in terms of permutations but this time for D_4:
(ABCD) -> (BADC) and so on.

I also understand that when n is greater than or equal to 3 that
dihedral groups are non-abelian, but do not have the time or
inclination to write the explanation for it.

I am wondering if I missing something in my understanding of these
groups or if my understanding thus far is incorrect.

Cheers,
Patrick

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