Abstract
Let G be a finite group, let p be a prime, and let P be 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 minimum possible number of factors of p-power order appearing in any normal series of G, in which each factor is either a p-group, a p’-group, or a non-p-solvable characteristically simple group, by the number of generators of P.
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We would like to thank the anonymous referees for useful comments on preliminary versions of the article.
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J. González-Sánchez was supported by grants MTM2011-28229- C02-01 and MTM2014-53810-C2-2-P, from the Spanish Ministry of Economy and Competitivity, the Ramon y Cajal Programme of the Spanish Ministry of Science and Innovation, grant RYC-2011-08885, and by the Basque Government, grants IT753-13 and IT974-16. J. Tent was supported by grants MTM2011-28229- C02-01 and MTM2014-53810-C2-2-P, from the Spanish Ministry of Economy and Competitivity, and by Prometeo II/Generalitat Valenciana.
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González-Sánchez, J., Tent, J. A cohomological criterion for p-solvability. Arch. Math. 111, 337–347 (2018). https://doi.org/10.1007/s00013-018-1229-2
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DOI: https://doi.org/10.1007/s00013-018-1229-2