My Math Forum ax+b group

 Real Analysis Real Analysis Math Forum

 October 20th, 2010, 10:31 AM #1 Newbie   Joined: Oct 2010 Posts: 17 Thanks: 0 ax+b group In "A Short Course on Spectral Theory", page 10, William Arveson asserts that the "ax+b group", ie. the group generated by all dilations and translations of the real line, is isomorpic to the group of all (real) 2x2 matrices of the form a b 0 1/a a>0, b real. It is very easy to check that the ax+b group is isomorphic to the group of all matrices of the form a b 0 1 a>0, b real. So these two matrix groups should be isomorphic. Is this correct, and if so, could someone please give me the isomorphism? I've tried for a while and can't seem to get it.
 October 20th, 2010, 12:13 PM #2 Global Moderator     Joined: Nov 2006 From: UTC -5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms Re: ax+b group Are these groups under + or *? Edit: Must be *. So the first takes (a, b) * (c, d) to (ac, ad+b/c) and the second takes (a, b) * (c, d) to (ac, ad + b).
 October 20th, 2010, 01:04 PM #3 Newbie   Joined: Oct 2010 Posts: 17 Thanks: 0 Re: ax+b group That's right. But how are these isomorphic?
October 20th, 2010, 03:14 PM   #4
Global Moderator

Joined: Nov 2006
From: UTC -5

Posts: 16,046
Thanks: 938

Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms
Re: ax+b group

Quote:
 Originally Posted by hytteteppe That's right. But how are these isomorphic?
Haven't figured it out.

 Tags group

### ax b group

Click on a term to search for related topics.
 Thread Tools Display Modes Linear Mode

 Similar Threads Thread Thread Starter Forum Replies Last Post Vasily Abstract Algebra 6 June 5th, 2012 02:58 PM PeterMarshall Abstract Algebra 2 January 24th, 2012 01:13 PM polysot Abstract Algebra 1 February 12th, 2011 05:31 PM sunflower Abstract Algebra 0 October 15th, 2010 01:20 PM mingcai6172 Real Analysis 0 March 21st, 2009 02:35 PM

 Contact - Home - Forums - Cryptocurrency Forum - Top