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October 20th, 2010, 10:31 AM   #1
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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.
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October 20th, 2010, 12:13 PM   #2
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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).
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October 20th, 2010, 01:04 PM   #3
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Re: ax+b group

That's right. But how are these isomorphic?
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October 20th, 2010, 03:14 PM   #4
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Re: ax+b group

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