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:
 

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