# Article

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Keywords:
principal block; $p$-radical group; $p$-radical block
Summary:
Let $G$ be a finite group and $k$ a field of characteristic $p > 0$. In this paper, we obtain several equivalent conditions to determine whether the principal block $B_{0}$ of a finite $p$-solvable group $G$ is $p$-radical, which means that $B_{0}$ has the property that $e_{0} (k_P)^G$ is semisimple as a $kG$-module, where $P$ is a Sylow $p$-subgroup of $G$, $k_{P}$ is the trivial $kP$-module, $(k_{P})^{G}$ is the induced module, and $e_{0}$ is the block idempotent of $B_{0}$. We also give the complete classification of a finite $p$-solvable group $G$ which has not more than three simple $B_{0}$-modules where $B_0$ is $p$-radical.
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