Principal blocks and $p$-radical groups
Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 431-444.

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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.
DOI : 10.1007/s10587-016-0266-x
Classification : 20C05, 20C20
Mots-clés : principal block; $p$-radical group; $p$-radical block 
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Hu, Xiaohan; Zeng, Jiwen. Principal blocks and $p$-radical groups. Czechoslovak Mathematical Journal, Tome 66 (2016) no. 2, pp. 431-444. doi : 10.1007/s10587-016-0266-x. https://geodesic-test.mathdoc.fr/articles/10.1007/s10587-016-0266-x/

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