Group exercise for rotations of regular n-gon objects

  • #1
LCSphysicist
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Homework Statement:
\
Relevant Equations:
.
1608184421413.png

The doubt is about B and C.

b)
n = 4, $C = {I,e^{2\pi/4}}
n = 5, $C = {I,e^{2\pi/5}}
n = 6, $C = {I,e^{2\pi/6}}

Is this right?

c)
I am not sure what does he wants...
 
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Answers and Replies

  • #2
Gaussian97
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Well, let's start by case ##n=4## in part b):
What you have is essentially
$$\langle e, r^2\rangle$$
But, from part a), if you did it correctly, you should know that ##r^4=e## and therefore
$$\langle e, r^2\rangle = \{e, r^2\} \neq C_4$$
So no, what you have is not a generating set.
The same is true for case ##n=6## where
$$\langle e, r^2\rangle = \{e, r^2, r^4\}\neq C_6$$
You did it correctly for ##n=5## where
$$\langle e, r^2\rangle = \{e, r^2, r^4, r, r^3\} = G_5$$

Now, you should first correct the cases ##n=4,6## and then, for case ##n=5## is this the only minimal generating set?
 
  • #3
LCSphysicist
634
153
Well, let's start by case ##n=4## in part b):
What you have is essentially
$$\langle e, r^2\rangle$$
But, from part a), if you did it correctly, you should know that ##r^4=e## and therefore
$$\langle e, r^2\rangle = \{e, r^2\} \neq C_4$$
So no, what you have is not a generating set.
The same is true for case ##n=6## where
$$\langle e, r^2\rangle = \{e, r^2, r^4\}\neq C_6$$
You did it correctly for ##n=5## where
$$\langle e, r^2\rangle = \{e, r^2, r^4, r, r^3\} = G_5$$

Now, you should first correct the cases ##n=4,6## and then, for case ##n=5## is this the only minimal generating set?
Hello. I think i do not understand yet the b. You said that what i wrote in the first and third case is essentially the same as $\langle e, r^2\rangle$, but my aim was to be the same as $\langle e, r\rangle$. Can't i call it equal r? So technically $r^4 = e$
 
  • #4
Gaussian97
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Hello. I think i do not understand yet the b. You said that what i wrote in the first and third case is essentially the same as $\langle e, r^2\rangle$, but my aim was to be the same as $\langle e, r\rangle$. Can't i call it equal r? So technically $r^4 = e$
Oh yes, sorry my mistake, I read the exponential of ##4\pi## instead of the ##2\pi##. Then yes what you wrote is just ##\langle e, r \rangle## and it works for ##C_4## and ##C_6##, but as I have shown you for ##C_5##, ##\langle e, r^2\rangle## is also a minimal generating set. So now the question is, are there more minimal generating sets? You need to find them all.
 
  • #5
haruspex
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I don't see why you have I in the generating sets. Isn't it redundant? Indeed, it mentions sets of size 2 for n=6, and I can see what they are, but your system would make them size 3.
Your list is incomplete for each of 4, 5, 6.
 

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