Proposition 9

Suppose that the field K has characteristic zero. Then the degree of every simple representation of G divides the cardinal of G.

Let $(V, \pi )$ be a simple representation of G and $\chi$ be its character. For every element $a$ of Z(K[G]), the endomorphism $\pi (a)$ of V is a homothety (VIII, p. 47, Theorem 1); denote by $\varphi (a)$ the scalar such that $\pi (a) =\varphi (a)_V$. The resulting mapping $\varphi$ from Z(K[G]) to K is an algebra homomorphism. Let us take $a=\sum_{g\in G}\chi (g^{-1})g$; by the remark of VIII, p. 408, we have $\varphi (a) =$ (dim $V$)$^{-1}|G|$. On the other hand, $a$ belongs to the subring $\mathscr{O}_n[G]\cap Z(K[G])$ of K[G], which is a finitely generated $\mathbf{Z}$-module (VII, §3, p. 15, Corollary). So the element $\varphi (a) =$ (dim $V$)$^{-1}|G|$ of K belongs to a subring of K that is a finitely generated $\mathbf{Z}$-module. We conclude using the following lemma.