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- 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 \(R_{180}\)

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- ganeshie8

- schrodinger

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- zzr0ck3r

https://ysharifi.wordpress.com/2011/02/02/center-of-dihedral-groups/

- zzr0ck3r

I can help with any questions.

- zzr0ck3r

if needed....

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- 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

I believe reflections have order \(2\) as two reflections does nothing
and rotations have order \(n\) as \(n*\dfrac{360}{n}=360\)

- zzr0ck3r

often it is the defining relation.

- ganeshie8

so in that proof, i assume
\(a\) = reflection
\(b\) = rotation

- zzr0ck3r

if it is not, you can get there from the rules of the group.

- zzr0ck3r

yeah, and order is |2n| (some people say D_4 had 8 elements) booo

- ganeshie8

\(ba=ab^{-1}\) works only for dihedral groups right ?
it shouldn't work for groups in general.. ?

- zzr0ck3r

correct

- zzr0ck3r

well, it might work for other, but it defined the dihedrals

- zzr0ck3r

you can also use \(b^{-1}a=ab\)

- 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

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

sure we can think of rotation and reflection as functions right

- 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

yeah so its aa bijection in two space where every pair of points maintains distance

- zzr0ck3r

and something way of saying that it stays in the same spot.

- zzr0ck3r

sorry it is late :)

- ganeshie8

I'm liking group theory but also feeling overwhelmed with all the different terminology and new stuff..

- ganeshie8

I think I understood the proof, thanks again! gnite !

- 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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