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

As G is a finite group

If m=n then we are done.

If

then there exists an element s.t.

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 the we have a contradiction?

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

If m=n then we are done.

what is n??
If

then there exists an element

s.t.

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

the we have a contradiction?

...

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....

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.

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

Re: symmetry of plane figures question

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.

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.

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!