The two groups $\mathbb R$ and $\mathbb C$ are isomorphic.
Solution: We know that $\mathbb {R(Q)}$ and $\mathbb {R^2(Q)}$ are isomorphic as vector spaces.
So,there is a vector space isomorphism between $\mathbb R$ and $\mathbb R^2$ over the field $\mathbb Q$,
Let it be $\phi$.
Then, for all $x,y\in \mathbb R$,we have $f(x+y)=f(x)+f(y)$ and $f$ is a bijection between $\mathbb R$ and $\mathbb R^2$.
We can replace $\mathbb R^2$ by $\mathbb C$ noting that there is a one-one correspondence $(x,y)\mapsto x+iy$ that preserves the operation.
So,$\mathbb {(R,+)}$ and $\mathbb {(C,+)}$ are isomorphic.
Is my solution correct?
