# Faithful representation and center of a group

In this topic we’ll solve following problem:

Let $\rho$ be an irreducible representation of $G$ of degree $n$ and character $\chi$; let $C$ be the center of $G$ and let $c$ be its order.

(a)Show that $\rho_s$ is a homothety for each $s\in C$. Deduce from this that $|\chi(s)|=n$ for all $s\in C$.

(b)Prove the inequality $n^2\leq \dfrac{g}{c}$.

(c)Show that, if $\rho$ is faithful, the group $C$ is cyclic.

The problem is posted in the website mathscope.org by PnAT and solved by 2M and Vodka, they are members of the site too. This is it in detail:

(a)For each $s\in C$ and $h\in G$ we have $\rho_h\rho_s=\rho_{hs}=\rho_{sh}=\rho_s\rho_h$. Therefore by Schur’s lemma we have $\rho_s$ is a homothety. Now for all $s\in C$ put $\rho_s=\lambda_sId_V$, since $\rho_{s^g}=Id_V$ we have $\lambda_s$ is a root of unit, particularly $|\lambda_s|=1$. Final, $|\chi(s)|=|Tr(\rho_s)|=|n\lambda_s|=n$.

(b)This part is easy! In fact, we have $g=\sum_{s\in G}|\chi(s)|^2\geq \sum_{s\in C}|\chi(s)|^2=\sum_{s\in C}n^2=cn^2$ and we’re done.

(c)If $\rho$ is faithful then by all what above we can see $C$ as a subgroup of the group $g-$th roots of unit, and therefore $C$ is cyclic.