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.
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.
|
