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September 3, 2013

Automorphism of the Additive Group of Rationals

Any automorphism of the group Q under addition is of the form x → qx for some q ∈ Q.


Discussion: Suppose f is an automorphism of the group Q. Let f(1) = m (of course 'm' will be different for different automorphisms). Now f(x+y) = f(x) + f(y) implies f(x) = mx where m is a constant and x belongs to set of integers (Cauchy's functional equation).

Now suppose x is rational. Then x = p/q where p and q are integers. Hence f(p) = mp. But p = qx hence f(p) = f(qx) = f(x+x+ ... + x) = qf(x)

There fore mp = qf(x)  implies m \times {\frac{p}{q} }= f(x) \implies f(x) = mx where m = f(1)

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