ganeshie8
  • ganeshie8
consider a regular \(2n-gon\) show that \(R_{180}\) commutes with all other rotations and flips. for example, below two sequence of transformations give the same final effect : 1) a. rotate 180 b. flip over the symmetry line passing through vertex A 2) a. flip over the symmetry line passing through vertex A b. rotate 180 in other words, show that the order doesn't matter if one transformation is \(R_{180}\)
Mathematics
schrodinger
  • schrodinger
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zzr0ck3r
  • zzr0ck3r
https://ysharifi.wordpress.com/2011/02/02/center-of-dihedral-groups/
zzr0ck3r
  • zzr0ck3r
I can help with any questions.
zzr0ck3r
  • zzr0ck3r
if needed....

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ganeshie8
  • ganeshie8
Thank you! that looks like a neat proof :) Im still going through the proof... Is there any intuitive way to convince ourselves that \(ba = ab^{-1}\) for dihedral groups ?
ganeshie8
  • ganeshie8
I believe reflections have order \(2\) as two reflections does nothing and rotations have order \(n\) as \(n*\dfrac{360}{n}=360\)
zzr0ck3r
  • zzr0ck3r
often it is the defining relation.
ganeshie8
  • ganeshie8
so in that proof, i assume \(a\) = reflection \(b\) = rotation
zzr0ck3r
  • zzr0ck3r
if it is not, you can get there from the rules of the group.
zzr0ck3r
  • zzr0ck3r
yeah, and order is |2n| (some people say D_4 had 8 elements) booo
ganeshie8
  • ganeshie8
\(ba=ab^{-1}\) works only for dihedral groups right ? it shouldn't work for groups in general.. ?
zzr0ck3r
  • zzr0ck3r
correct
zzr0ck3r
  • zzr0ck3r
well, it might work for other, but it defined the dihedrals
zzr0ck3r
  • zzr0ck3r
you can also use \(b^{-1}a=ab\)
ganeshie8
  • ganeshie8
I see that rotation composed with reflection changes the orientation, so it is essentially a reflection. Since reflection is its own inverse, we have : \[(rotation)(reflection) \\= ((rotation)(reflection))^{-1} \\= (reflection)^{-1}(rotation)^{-1} \\=(reflection)(rotation)^{-1}\] that convinces me but not sure if it is a valid proof.. .
zzr0ck3r
  • zzr0ck3r
A valid proof (in my eyes) would need to be a proof in terms of functions, and then you must define what a symmetry is in terms of function, which is strange enough and often skipped, and essentially what you will be doing is creating group theory.
ganeshie8
  • ganeshie8
sure we can think of rotation and reflection as functions right
zzr0ck3r
  • zzr0ck3r
This is exactly how we start our class in group theory. 1) hand them triangles. 2) give them hints untill they can sort of define what a symmetry is. 3) let them play around with names and notation 4) give them hints until they figure out that relation hehe
zzr0ck3r
  • zzr0ck3r
yeah so its aa bijection in two space where every pair of points maintains distance
zzr0ck3r
  • zzr0ck3r
and something way of saying that it stays in the same spot.
zzr0ck3r
  • zzr0ck3r
sorry it is late :)
ganeshie8
  • ganeshie8
I'm liking group theory but also feeling overwhelmed with all the different terminology and new stuff..
ganeshie8
  • ganeshie8
I think I understood the proof, thanks again! gnite !
zzr0ck3r
  • zzr0ck3r
Yeah it gets better :) So much notation and definitions. But like with the rest of this stuff, once you start to figure out why they do it to begin with, that other stuff will seem more natural:)

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