每日定理14

Isaacs, $\textit{Character Theory of Finite Groups}$, Corollary(2.6)

The group $G$ is abelian iff every irreducible character is linear.

Pf:

• The number  $k$ of classes of $G$ equals $|G|$ iff $G$ is abelian.
• $|G|=\sum_{i=1}^k\chi_i(1)^2$. 

Isaacs, $\textit{Character Theory of Finite Groups}$, Corollary(2.7) 

Let $G$ be a group. Then $|Irr(G)|$ equals the number of conjugacy classes of $G$ and

$$\sum_{\chi\in Irr(G)}\chi(1)^2=|G|.$$ 

Pf:

• To show that the $\chi_i$ are distinct.
• $\chi_i(e_j)=0$ if $i\neq j$ and $\chi_i(e_i)=\chi_i(1)\neq0$.