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 $O_{p'}(G)\leq O_{p'p}(G)=HO_{p'}(G)= \textrm{Ker}(B_{0}^{*}) \leq O_{p'pp'}=G$, where $H=G^{'}O^{p'}(G), \textrm{Ker}(B_{0}^{*})=\bigcap_{\lambda \in \textrm{IBr}(B_{0})} \textrm{Ker}(V_{\lambda}), B_{0}$ 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 $$ \frac{\textrm{Dim}{F}(B)}{|D|}= \sum{\phi \in \textrm{IBr}(B)}\phi(1)^{2}$$ where $D$ is the defect group of $B$.