At 01:14 PM 5/9/03 -0400, Michael Kleber wrote:
Ray Tayek asked:
hi, iirc, the elements are : e, r, r2, r3, h, v, d1, and d2.
does h flip about the x or y axis? ...
Those names seem pretty mnemonic to me -- I presume that your 'h' and 'v' flip over the 'horizontal' and 'vertical' axes.
my h flipped over the y axis - maybe a bad choice ( i was using '|' on the screen though).
Of course, it doesn't matter which diagonal is called d1 and which is d2, it's all up to the person deciding to name them!
To be as clear as possible, you could always name them
id, rot(90), rot(180), rot(270), refl(/), refl(\), refl(-), refl(|).
yes, that's what i am doing.
I think that's completely unambiguous.
Or I did, until Eugene Salamin wrote:
The dihedral groups contain only proper rotations, no reflections or inversions. D4 has 8 elements. It has the 4-element subgroup C4 of rotations about the z-axis, and four 180-degree rotations about axes that lie in the xy-plane and are separated by 45 degrees.
excellent point.
Let's try to be clear here. "The dihedral group of order 8" is an abstract group of 8 elements, one of only two nonabelian such. ....
There's also an action of the same group on 3-dimensional space, in which you only need rotations: things in the xy plane work just as in two dimensions (so the above is a subrepresentation), and anything that looked like a reflection also takes the z-axis to its negative, so it becomes a product of two reflections, which is a rotation. If you take a physical square and look at all of its symmetries, this is what you're doing: ...
To a representation theorist, the 3-dim rotation rep is "decomposable" as a sum of the 2-dim rep on the fixed xy plane plus a 1-dim rep on the z-axis, taking the rotations to +1 and the reflections to -1. ..
this clears it all up for me. thanks people --- ray tayek http://tayek.com/ actively seeking mentoring or telecommuting work vice chair orange county java users group http://www.ocjug.org/ hate spam? http://samspade.org/ssw/