# Thread: symmetry of plane figures question

1. ## symmetry of plane figures question

Let G be a finite group of rotations of the plane around the origin, prove that G is cyclic.

Im confused about how to start this (I have not done any group theory in a while)

Thanks for any help

Take the smallest rotation, denoted as g, then there exists a least m such that $g^m=1$

As G is a finite group $|G|=n$

If m=n then we are done. $\{1,g,g^2,.....,g^{m-1}\}$

If $m

then there exists an element $b\in G$ s.t. $b\neq g^i$

do I then go on to show that if this was the case b must have an inverse and that would be one of the $g^i's$ the we have a contradiction?

2. ## Re: symmetry of plane figures question

Originally Posted by hmmmm
Let G be a finite group of rotations of the plane around the origin, prove that G is cyclic.

Im confused about how to start this (I have not done any group theory in a while)

Thanks for any help

Take the smallest rotation, denoted as g, then there exists a least m such that $g^m=1$

If m=n then we are done. $\{1,g,g^2,.....,g^{m-1}\}$
what is n??
If $m

then there exists an element $b\in G$ s.t. $b\neq g^i$

do I then go on to show that if this was the case b must have an inverse and that would be one of the $g^i's$ the we have a contradiction?
...

3. ## Re: symmetry of plane figures question

oops sorry it is the order of the group G, but i think what I have done may be wrong....

4. ## Re: symmetry of plane figures question

Originally Posted by hmmmm
oops sorry it is the order of the group G, but i think what I have done may be wrong....
well if that's what n is then m has to be equal to n because you said that g represents the smallest rotation.

5. ## Re: symmetry of plane figures question

Im sorry im a bit confused by what you have said, if n=m then I am done, could you possibly give more explanation?

thanks for any help

6. ## Re: symmetry of plane figures question

Originally Posted by hmmmm
Let G be a finite group of rotations of the plane around the origin, prove that G is cyclic.

Im confused about how to start this (I have not done any group theory in a while)

Thanks for any help

Take the smallest rotation, denoted as g, then there exists a least m such that $g^m=1$

As G is a finite group $|G|=n$

If m=n then we are done. $\{1,g,g^2,.....,g^{m-1}\}$

If $m

then there exists an element $b\in G$ s.t. $b\neq g^i$

do I then go on to show that if this was the case b must have an inverse and that would be one of the $g^i's$ the we have a contradiction?
I assume you know that every rotation of the plane is of the form $\begin{pmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{pmatrix}$. It's easy then to see that the map $\begin{pmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{pmatrix}\mapsto e^{i\theta}$ defines an embedding (injective homomorphism) $G\hookrightarrow \mathbb{C}^\times$. Thus, it suffices to prove that every finite subgroup $H$ of $\mathbb{C}^\times$ is cyclic. To do this we merely note that by Lagrange's theorem $x^{|H|}-1=0$ for all $x\in H$ and by the FTA we know that $x^{|H|}-1$ has precisely $|H|$ roots, and moreover we know they are the $|H|$ roots of unity which are trivially cyclic.

7. ## Re: symmetry of plane figures question

Thanks, not the way that I think I was meant to do this but I think I get it (this seems a little more complicated then what I was expecting).

Thanks very much.

8. ## Re: symmetry of plane figures question

Originally Posted by hmmmm
Thanks, not the way that I think I was meant to do this but I think I get it (this seems a little more complicated then what I was expecting).

Thanks very much.
You can do the minimum angle whosuwutsit if you want or you can try doing a similar proof to mine except using characteristic or minimal polynomials.

9. ## Re: symmetry of plane figures question

no your proof is much nicer than what I was attempting to do. all I was saying was that I had not thought of doing that at all. thanks very much!