Find all the group homomorphisms from $(\mathbb{Q}, +)$ into $(\mathbb{R}, +)$.

My attempt:

If $\mathbb{Q}$ were a cyclic group, I could tell that any homomorphism will be determined by the image of generator. But here $\mathbb{Q}$ is not a cyclic group, so there's no generator. All one can say is that:

  1. if $f$ is a homomorphism then $f(0)=0$.

But this doesn't help me to solve this problem. So how should it be tackled?

  • $\begingroup$ What are $ Q$ and $R$? $\endgroup$ – Daniel Aug 17 '15 at 16:55
  • $\begingroup$ additive group of rationals and additive group of all reals, i think. $\endgroup$ – Foggy Aug 17 '15 at 16:57
  • $\begingroup$ I suppose $Q$ and $R$ mean Rational and Real numbers. The problem is whether those are seen as groups or fields or something else. $\endgroup$ – Jose Paternina Aug 17 '15 at 16:59
  • $\begingroup$ @JosePaternina OP said in the question he/she is looking for group homomorphisms. $\endgroup$ – Silvia Ghinassi Aug 17 '15 at 17:01
  • $\begingroup$ @SilviaGhinassi you're right, sorry. $\endgroup$ – Jose Paternina Aug 17 '15 at 17:06

Let $f: (\mathbb{Q}, +) \to (\mathbb{R}, +)$ be a group homomorphism. As you say, $f(0) = 0$.

What happens to $1$? Let's say $f(1) = x$. Now, this fixes all the naturals: $f(1+1) = f(1) + f(1) = 2x$, and so on, so $f(n) = nx$.

What happens to $\frac{1}{2}$, which is what comes to mind as the simplest non-integer rational? $x = f(\frac{1}{2} + \frac{1}{2}) = 2 f(\frac{1}{2})$, so $f(\frac{1}{2}) = \frac{x}{2}$.

Can you generalise that yourself?

  • $\begingroup$ let m/n be any rational, i think then f(n(m/n))=nf(m/n), this means f(m/n)=(mx)/n..? is that all so we determined one morphism by defining both naturals and rationals.. so, is there only one morphisms between these two groups? $\endgroup$ – Foggy Aug 17 '15 at 17:09
  • $\begingroup$ You're right in saying that $f(n \times \frac{m}{n}) = n f(\frac{m}{n})$, and so $f(\frac{m}{n}) = \frac{m x}{n}$. That means there is only one morphism after having fixed the image of $1 \in \mathbb{Q}$. How many morphisms does that mean there are in total between the two groups? $\endgroup$ – Patrick Stevens Aug 17 '15 at 17:25
  • $\begingroup$ @PatrickStevens do you mean that there are c number of morphisms between the groups? $\endgroup$ – reflexive Feb 6 '18 at 4:34
  • $\begingroup$ @ThatIs Correct. $\endgroup$ – Patrick Stevens Feb 6 '18 at 8:38

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