Rationality of characters

Question

Let $G$ be a group. I would like to show TFAE:
$1.$ Every irreducible character of $G$ is rational-valued;
$2.$ For $g,h \in G$, $g,h$ are conjugate in $G$;
$3.$ The cyclic groups $\langle g \rangle$ and $\langle h \rangle$ are conjugate in $G$.
I have proved $1 \implies 2$ by considering the Galois extension $\mathbb{Q}(\omega)/\mathbb{Q}$ for $\omega$ an root of unity and showing that if the character is rational, it is fixed by an element of the Galois group of this, so $\chi(h)=\sigma(\chi(g))=\chi(g)$, where $\sigma \in Gal(\mathbb{Q}(\omega)/\mathbb{Q})$, so $g$ and $h$ must be conjugate.
However, I am not sure how to show $2 \implies 3$ and $3 \implies 1$. There is a section on this in Serre's book "Linear Representations of Finite Groups" (Ch. 13), but he does not give a proof.