A coprime action version of a solubility criterion of Deskins
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Let A and G be finite groups of relatively prime orders and suppose that A acts on G via automorphisms. We demonstrate that if G has a maximal A-invariant subgroup M that is nilpotent and the Sylow 2-subgroup of M has class at most 2, then G is soluble. This result extends, in the context of coprime action, a solubility criterion given by W.E. Deskins.
KeywordsSoluble groups Maximal subgroups Coprime action Group action on groups
Mathematics Subject Classification20D20 20D15
The first author is partially supported by the Valencian Government, Proyecto PROMETEOII/2015/011 and also by Universitat Jaume I, Grant P11B-2015-77. The second author is supported by the NNSF of China (No. 11301218) and the Nature Science Fund of Shandong Province (No. ZR2014AM020).
- 1.Beltrán, A., Shao, C.G.: Restrictions on maximal invariant subgroups implying solvability of finite groups. Ann. Mat. Pura Appl. (2018). https://doi.org/10.1007/s10231-018-0777-1
- 7.Kondrat’ev, A.S.: Normalizers of the Sylow 2-subgroups in finite simple groups. Math. Zametki 78(3), 368–376 (2005); translation in Math. Notes 78(3–4), 338–346 (2005) (Russian) Google Scholar