Group exercise for rotations of regular n-gon objects

In summary, the conversation is about finding the generating sets for the cyclic groups C4, C5, and C6. It is determined that for n=4 and n=6, the set containing e and r is a generating set, while for n=5, the set containing e, r^2, r^4, r, and r^3 is a generating set. It is then questioned if there are any other minimal generating sets. Additionally, there is a discussion about whether I should be included in the generating sets, with the conclusion that it is redundant. The list of generating sets provided is incomplete for each of the values 4, 5, and 6.
  • #1
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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...
 
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  • #2
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?
 
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  • #3
Gaussian97 said:
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
Herculi said:
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
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.
 

1. What is group exercise for rotations of regular n-gon objects?

Group exercise for rotations of regular n-gon objects is a mathematical concept that involves manipulating and transforming regular polygons (shapes with equal sides and angles) by rotating them around a center point. This exercise helps to understand the properties and symmetries of regular polygons.

2. Why is it important to learn about group exercise for rotations of regular n-gon objects?

Learning about group exercise for rotations of regular n-gon objects helps to develop spatial reasoning skills and a deeper understanding of geometry. It also has practical applications in fields such as architecture, engineering, and computer graphics.

3. How do you perform group exercise for rotations of regular n-gon objects?

To perform group exercise for rotations of regular n-gon objects, you need to choose a regular polygon and a center point. Then, rotate the polygon around the center point in a clockwise or counterclockwise direction by a certain angle. Repeat this process multiple times to observe the resulting transformations.

4. What are some examples of group exercise for rotations of regular n-gon objects?

Some examples of group exercise for rotations of regular n-gon objects include rotating a square, triangle, or hexagon around its center point. You can also combine multiple rotations to create more complex transformations.

5. How does group exercise for rotations of regular n-gon objects relate to other mathematical concepts?

Group exercise for rotations of regular n-gon objects is closely related to other mathematical concepts such as symmetry, congruence, and transformational geometry. It also has connections to group theory, which is a branch of abstract algebra.

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