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.

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