Multiplicative groups of nonzero rational and real numbers: Isomorphic or not?

Question

Are the multiplicative groups $Q^*$ and $R^*$ isomorphic or not?

I have seen some solution talking about cardinality. How is cardinality related to whether they are isomorphic?

Answer 1

An isomorphism is a bijection. If there is a bijection between two sets, they have the same cardinality.

So, therefore, isomorphic objects (such as multiplicative groups, here) always have the same cardinality. And the contrapositive: objects with different cardinality cannot be isomorphic.

Answer 2

They are not isomorphic. Suppose $f$ is an isomorfism from $\mathbb R^*$ to $\mathbb Q^*$.

Let $a$ be such that $f(a)=2$. There exists $b$ such that $b^3=a$. It follows $f(b)^3=f(b^3)=f(a)=2$. Contradiction.