Trying to understand D_n in two dimensions - sci.math

Message from discussion Trying to understand D_n in two dimensions

Jim Heckman reply-to

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More options Nov 9 2009, 10:10 am

Newsgroups: sci.math

From: "Jim Heckman" <rot13(reply-to)@none.invalid>

Date: Mon, 9 Nov 2009 10:10:08 GMT

Local: Mon, Nov 9 2009 10:10 am

Subject: Re: Trying to understand D_n in two dimensions

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On 7-Nov-2009, eratosthenes <rehamkcir...@gmail.com>
wrote in message
<195d5518-8948-4ae2-a216-fedc36838...@37g2000yqm.googlegroups.com>:

> 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

Careful, n isn't the order of the dihedral group, but rather of its
cyclic subgroup of index 2.

> 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.

Hmm... There are n/2 "flip" axes through opposite vertices, and
another n/2 through the centers of opposite sides (what I'm
assuming you're calling "flats"), for a total of n flip axes, just
as when n is odd. The difference is that when n is odd, each flip
axis runs through a vertex and the center of the opposite side. In
all cases, the flip moves all vertices except the 0, 1 or 2 on the
axis, and does the same for the centers of the sides.

> 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.

Calculate what happens when you do a rotation of 2pi/n followed by
a flip, versus doing the same flip first, followed by the 2pi/n
rotation. In both cases the result is a simple flip, but the two
flips are along different axes. Can you see what the relationship
is between the two axes, as a function of n? Be sure to check both
types of flip when n is even.

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

--
Jim Heckman

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