A cohomological criterion for $p$-solvability

Let $G$ be a finite group, $p$ a prime and $P$ a Sylow $p$-subgroup of $G$. In this note we give a cohomological criterion for the $p$-solvability of $G$ depending on the cohomology in degree $1$ with coefficients in $\mathbb F_p$ of both the normal subgroups of $G$ and $P$. As a byproduct we bound the minimal number of quotients of order a power of $p$ appearing in any normal series of $G$ by the number of generators of $P$.

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