A cohomological criterion for $p$-solvability

TL;DR

This paper introduces a cohomological criterion based on degree 1 cohomology with coefficients in _p for determining the p-solvability of finite groups, linking subgroup cohomology to group structure.

Contribution

It provides a new cohomological criterion for p-solvability of finite groups based on the cohomology of normal subgroups and Sylow p-subgroups.

Findings

Cohomological criterion for p-solvability established

Bound on the minimal number of p-power order quotients in a normal series

Relation between the number of generators of P and the structure of G

Abstract

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 $P$.