Isomorphic Groups

Date: 02/11/2003 at 19:16:37
From: Scott
Subject: Isomorphic groups

Is the additive group of rationals isomorphic to the multiplicative 
group of non-zero rationals?

TO prove isomorphism I must find a bijective map such that f(xy) = f
(x)f(y). However, I can't find one that will map reals to real and 
preserve that 0 must get mapped to 1. I think that they are not 
isomorphic but cannot locate anything in their structure that is 
different.

Date: 02/12/2003 at 02:56:37
From: Doctor Jacques
Subject: Re: Isomorphic groups

Hi Scott,

Assume that there is such an isomorphism. Then there is a rational q 
such that f(q) = 2.

Now, q/2 is also rational, and we should have:

  f(q/2 + q/2) = f(q/2)*f(q/2) = 2

which implies that f(q/2) = sqrt (2) or -sqrt(2). As neither of these 
is rational, we reach a contradiction - there is no such isomorphism.

Note:

I could also have used -1 instead of 2, but this proves that the 
additive group is not even isomorphic to the multiplicative group of 
non-zero _positive_ rationals. If you replace "rationals" by "reals" 
in this last case, there is such an isomorphism, namely the 
exponential function.

- Doctor Jacques, The Math Forum
  http://mathforum.org/dr.math/