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.
SRE实战 互联网时代守护先锋，助力企业售后服务体系运筹帷幄！一键直达领取阿里云限量特价优惠。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 $\chi_i(e_i)=\chi_i(1)\neq0$.
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