每日定理4

Posted 2021-02-17 zhengtao1992

Isaacs, $ extit{Character Theory of Finite Groups}$, Problems(1.9)

Let $G$ be a group and $F$ a field of characteristic $p$. Suppose $pmid|G|$, then $F[G]$ is not semisimple.

Pf: Consider regular module $F[G]^{circ}$ and prove by contradiction.

• $0 eq v=sum_{gin G}gin Z(F[G])$
• $F[G]^{circ}=langle v angleoplus W$, $langle v angle={va|ain F[G]}$
• $v=1v=(va+w)v=vav+wv=v^2a+wv=wvin W$