Group exercise for rotations of regular n-gon objects

[LCSphysicist]

Homework Statement

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Relevant Equations

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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...

 

[Gaussian97]

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?

 

[LCSphysicist]

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 $r^4 = e$

 

[Gaussian97]

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.

 

[haruspex]

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.