Indices of Maximal Invariant Subgroups and Solvability of Finite Groups
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Let A and G be finite groups and suppose that A acts coprimely on G via automorphisms. We study the solvability and supersolvability of G when certain proper maximal A-invariant subgroups of G have prime index or when they have certain prime power indices in G.
KeywordsFinite groups maximal subgroups subgroup index coprime action group action on groups
Mathematics Subject Classification20D20 20D15
Part of this work was written while the first author was doing a research stay during 2017 at the Universidad Jaume I of Castellón, Spain. He would like to thank the Mathematics Department of the institution and A. Beltrán for their hospitality. The first author is supported by the NNSF of China (No. 11301218), the Nature Science Fund of Shandong Province (No. ZR2018AM020) and the Opening Project of Sichuan Province University Key Laboratory of Bridge Non-destruction Detecting and Engineering Computing (No. 2018QZJ04). The second author was partially supported by the Valencian Government, Proyecto PROMETEOII/2015/011 and also by Universitat Jaume I, Grant P11B-2015-77.