# symmetry of plane figures question

• Jul 11th 2011, 06:35 AM
hmmmm
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 $\displaystyle g^m=1$

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

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

If $\displaystyle m<n$

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

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 $\displaystyle g^i's$ the we have a contradiction?
• Jul 11th 2011, 07:43 AM
abhishekkgp
Re: symmetry of plane figures question
Quote:

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 $\displaystyle g^m=1$

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

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

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 $\displaystyle g^i's$ the we have a contradiction?

...
• Jul 11th 2011, 07:45 AM
hmmmm
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....
• Jul 11th 2011, 08:08 AM
abhishekkgp
Re: symmetry of plane figures question
Quote:

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.
• Jul 11th 2011, 08:23 AM
hmmmm
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
• Jul 11th 2011, 08:56 AM
Drexel28
Re: symmetry of plane figures question
Quote:

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 $\displaystyle g^m=1$

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

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

If $\displaystyle m<n$

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

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 $\displaystyle g^i's$ the we have a contradiction?

I assume you know that every rotation of the plane is of the form $\displaystyle \begin{pmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{pmatrix}$. It's easy then to see that the map $\displaystyle \begin{pmatrix}\cos(\theta) & \sin(\theta)\\ -\sin(\theta) & \cos(\theta)\end{pmatrix}\mapsto e^{i\theta}$ defines an embedding (injective homomorphism) $\displaystyle G\hookrightarrow \mathbb{C}^\times$. Thus, it suffices to prove that every finite subgroup $\displaystyle H$ of $\displaystyle \mathbb{C}^\times$ is cyclic. To do this we merely note that by Lagrange's theorem $\displaystyle x^{|H|}-1=0$ for all $\displaystyle x\in H$ and by the FTA we know that $\displaystyle x^{|H|}-1$ has precisely $\displaystyle |H|$ roots, and moreover we know they are the $\displaystyle |H|$ roots of unity which are trivially cyclic.
• Jul 11th 2011, 09:19 AM
hmmmm
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.
• Jul 11th 2011, 09:20 AM
Drexel28
Re: symmetry of plane figures question
Quote:

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.
• Jul 11th 2011, 10:11 AM
hmmmm
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!