The problem involves proving that a finite group of rotations of the plane about the origin is cyclic. The original poster attempts to understand the implications of cyclic groups and how to demonstrate that all elements can be expressed as powers of a single element.
The discussion is ongoing, with participants exploring various lines of reasoning and attempting to clarify their understanding of the problem. Some guidance has been offered regarding the implications of the smallest rotation and the potential contradiction arising from the assumptions made about the group.
There is a focus on the properties of finite groups and the implications of having a smallest element in the context of rotations. Participants express uncertainty about how to resolve contradictions and the implications of their findings on the cyclic nature of the group.
Dick said:There are lots of ways to prove this depending on what tools you happen to have around. The group must have a smallest nonzero rotation r. All multiples of r must also be in the group. That makes a nice cyclic subgroup, right? If it's noncyclic there must be an element s which is not a multiple of r. Now what?
SNOOTCHIEBOOCHEE said:By division algorithm s= qr+ m . I am guessing we have to somehow show that m=0? Or show that if s is in a group then r is not.
dont know how to do that.