Linear characters and block algebra

Abstract

This paper will prove that: 1. $G$ has a block only having linear ordinary characters if and only if $G$ is a $p$-nilpotent group with an abelian Sylow $p$-subgroup; 2. $G$ has a block only having linear Brauer characters if and only if $O_{p^{\prime }}(G)\le O_{p^{\prime }p}(G)=H{O}_{p^{\prime }}(G)=\text{Ker}(B_0^{\ast })\le O_{p^{\prime }p{p}^{\prime }}=G$, where $H=G^{{}^{\prime }}{O}^{p^{\prime }}(G),\text{Ker}(B_0^{\ast })={\bigcap }_{\lambda \in \text{IBr}(B_0)}\text{Ker}(V_{\lambda }),B_0$ is the principal block of $G$ and $V_{\lambda }$ is the $F[G]$-module affording the Brauer character $\lambda$; 3. if $G$ satisfies the conditions above, then for any block algebra $B$ of $G$, we have $$\frac{{\text{Dim}}_F(B)}{|D|}=\sum _{\varphi \in \text{IBr}(B)}\varphi (1{)}^2$$ where $D$ is the defect group of $B$.

arXiv commentWe clear describe finite groups if it has a block algebra having only linear irreducible ordinary characters or irreducible Brauer characters

References in corpus (1)

• Decomposition of Cartan Matrix and conjectures on Brauer character degrees