This is a problem from S.-T. Yau College Student Mathematics Contests 2019.
Let $G$ be a finite group. Assume that for any representation $V$ of $G$ over a field of characteristic zero, the character $\chi_{V}$ takes value in $\mathbb{Q}$. Assume $g$ is an element in $G$ such that $g^{2019}=1$. Prove that $g$ and $g^{19}$ are conjugate in $G$.
Here is what I have got:
- Since $\chi_V$ is an algebraic integer, we can deduce $\chi_V$ takes value in $\mathbb{Z}$.
- To prove $g$ and $g^{19}$ are conjugate in $G$, it suffices to show $\chi_V(g)=\chi_V(g^{19})$ for any irreducible representation $V$, because any class functions of $G$ is a linear combination of $\chi_V$ where $V$ is irreducible.
I know the fact that $\chi_V$ takes value in $\mathbb{Z}$ plays an important role in this problem. But I don't know how to use it. I hope someone may help me. Thanks a lot!
$\DeclareMathOperator{\Gal}{Gal}$$\newcommand{\Q}{\mathbb{Q}}$Let $\omega$ be a $2019$-th primitive root of $1$. Since $19$ and $2019$ are coprime, $\omega^{19}$ is also such a primitive root, so that $\omega \mapsto \omega^{19}$ yields an element $\alpha \in \Gal(\Q(\omega)/\Q)$.
Fix an irreducible character $\chi$ of $G$, and let $\rho$ be a corresponding representation. Then we know that, with respect to a suitable basis, $\rho(g)$ is a diagonal matrix, with diagonal entries $\omega^{a_{1}}, \omega^{a_{2}}, \dots$, for suitable $a_{1}, a_{2}, \dots$.
Thus $\rho(g^{19}) = \rho(g)^{19}$ is also diagonal, with diagonal entries $\omega^{19 a_{1}}, \omega^{19 a_{2}}, \dots$
It follows that $$ \chi(g) = \omega^{a_{1}} + \omega^{a_{2}} + \dots, $$ and $$ \chi(g^{19}) = \omega^{19 a_{1}} + \omega^{19 a_{2}} + \dots, $$ so that $\chi(g^{19}) = \chi(g)^{\alpha} = \chi(g)$, as $\chi(g) \in \Q$.