mersenneforum.org  

Go Back   mersenneforum.org > New To GIMPS? Start Here! > Homework Help

Reply
 
Thread Tools
Old 2010-08-12, 03:26   #1
Unregistered
 

87510 Posts
Default Two Questions about Unit Circle

These aren't actually homework questions, but rather specific examples that came to mind during my reading. Obviously, I'm interested in the proof; not a Yes/No answer.

Let C be the unit circle in the complex plane. Regard it as a topological group, and a measure space.

1) Does C have closed infinite subgroups other than C itself?

2) Does C have subgroups of positive measure other than C itself?

Thanks
  Reply With Quote
Old 2010-08-12, 23:15   #2
Orgasmic Troll
Cranksta Rap Ayatollah
 
Orgasmic Troll's Avatar
 
Jul 2003

64110 Posts
Default

Are you considering the group C to just be rotations, or the set of all bicontinuous functions on the unit circle?
Orgasmic Troll is offline   Reply With Quote
Old 2010-08-13, 01:10   #3
Unregistered
 

2×709 Posts
Default

C = {e^{it}: 0\leq t<2\pi}
  Reply With Quote
Old 2010-08-13, 13:12   #4
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

1D2416 Posts
Default

Quote:
Originally Posted by Unregistered View Post
C = {e^{it}: 0\leq t<2\pi}
This does not answer the previous question.

I will offer a hint. If working in the group of rotations, what is the order
of each element in any purported infinite sub-group? Then ask : Can
this sub-group have finite index?
R.D. Silverman is offline   Reply With Quote
Old 2010-08-13, 18:58   #5
R.D. Silverman
 
R.D. Silverman's Avatar
 
Nov 2003

22×5×373 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
This does not answer the previous question.

I will offer a hint. If working in the group of rotations, what is the order
of each element in any purported infinite sub-group? Then ask : Can
this sub-group have finite index?
Also, I hope that you are not looking to prove Cartan's Theorem??????
R.D. Silverman is offline   Reply With Quote
Old 2010-08-15, 14:28   #6
Unregistered
 

4,139 Posts
Default

Quote:
Originally Posted by R.D. Silverman View Post
This does not answer the previous question.

I will offer a hint. If working in the group of rotations, what is the order
of each element in any purported infinite sub-group? Then ask : Can
this sub-group have finite index?
Sorry, I should have said complex multiplication is the operation.

Come to think of it, I read a theorem in Stein and Shakarchi's "Fourier Analysis" a few years ago that essentially said: If \theta\in [0,2\pi) is not a rational multiple of 2\pi, then the subgroup generated by e^{i\theta} is dense.

So if A\subset C is an closed proper subgroup, it can contain only points of the form e^{i\frac{2\pi k}{n}}. If e^{i\frac{2\pi k}{n}}\in A and we assume WLOG that gcd(k,n)=1, then A contains every n^{th} root of unity. If A is infinite, then it contains all n^{th} roots of unity for infinitely many n. Since any point in C is within a distance \frac{2\pi}{n} of some n^{th} root of unity, A is dense and hence equal to all of C.

Thank you
  Reply With Quote
Old 2010-08-15, 14:43   #7
Unregistered
 

2,083 Posts
Default

I got the second problem too by the way. It follows from an exercise I just completed today:

If G is a Polish locally compact group and equipped with its Haar measure and A\subset G has positive measure, than A^{-1}A contains an open neighborhood of 1.

But when A is a subgroup, A^{-1}A=A. So A contains an open neighborhood U of 1. Since any element of C is of the form u^n for some u\in U,n\geq 0, A=C.
  Reply With Quote
Reply

Thread Tools


Similar Threads
Thread Thread Starter Forum Replies Last Post
the unit circle, tangens and the complex plane bhelmes Math 0 2018-03-04 18:50
A hexagon inscribed in a circle Batalov Puzzles 9 2010-06-02 23:46
How to divide a circle into 16 pieces with only 5 lines???? 4 lil pigs mom Homework Help 52 2006-12-05 08:41
Circle intersection points Mini-Geek Puzzles 1 2006-11-27 03:20
Circle Puzzles 1 mfgoode Puzzles 18 2005-07-11 11:51

All times are UTC. The time now is 12:48.

Wed Dec 2 12:48:36 UTC 2020 up 83 days, 9:59, 3 users, load averages: 4.20, 4.56, 4.44

Powered by vBulletin® Version 3.8.11
Copyright ©2000 - 2020, Jelsoft Enterprises Ltd.

This forum has received and complied with 0 (zero) government requests for information.

Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or any later version published by the Free Software Foundation.
A copy of the license is included in the FAQ.