Abstract
Recall a result due to O. J. Schmidt that a finite group whose proper subgroups are nilpotent is soluble. The present note extends this result and shows that if all non-normal maximal subgroups of a finite group are nilpotent, then (i) it is soluble; (ii) it is p-nilpotent for some prime p; (iii) if it is not nilpotent, then the number of prime divisors contained in its order is between 2 and k + 2, where k is the number of normal maximal subgroups which are not nilpotent.
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References
Ballester-Bolinches A., Guo X.: Some results on p-nilpotence and solubility of finite groups. J. Algebra 228, 491–496 (2000)
Guo X., Shum K.P.: p-nilpotence of finite groups and minimal subgroups. J. Algebra 207, 459–470 (2003)
Huppert B.: Endliche Gruppen I. Springer-Verlag, Berlin/Heidelberg/New York (1983)
Robinson D.J.: A course in the theory of groups, 2nd ed. Springer-Verlag, New York/Heidelberg/Berlin (1996)
Shi J.: A note on p-nilpotence of finite groups. J. Algebra 241, 435–436 (2001)
Shi J., Shi W., Zhang C.: A note on p-nilpotence and solvability of finite groups. J. Algebra 321, 1555–1560 (2009)
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This work was supported by the National Natural Science Foundation of China (Grant No. 10771132), the Natural Science Foundation of the Ministry of Education of China for the Returned Overseas Scholars (Grant No. 2008101) and the Natural Science Foundation of Shanxi for the Returned Overseas Scholars (Grant No. 200799).
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Li, Q., Guo, X. On p-nilpotence and solubility of groups. Arch. Math. 96, 1–7 (2011). https://doi.org/10.1007/s00013-010-0215-0
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DOI: https://doi.org/10.1007/s00013-010-0215-0