Two Theorems about Nilpotent Subgroup

Lijiang Zeng

.

DOI: 10.4236/am.2011.25074                

Abstract

In the paper, we introduce some concepts and notations of Hall π-subgroup etc, and prove some properties about finite p-group, nilpotent group and Sylow p-subgroup. Finally, we have proved two interesting theorems about nilpotent subgroup.

Keywords

Hall π-Subgroup, Sylow p-Subgroup, Normalizer, Nilpotent Group

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	M. Osima, “On the Induced Characters of a Group,” Proceedings of the Japan Academy, Vol. 28, No. 5, 1952, pp. 243-248. doi:10.3792/pja/1195570968
[2]	J. D. Dixou, “The Structure of Linear Groups,” Van Nostrand Reinhold, New York, 1971, pp. 189-201.
[3]	M. Aschbacher, “Finite Group Theory,” Cambridge University Press, New York, 1986, pp. 174-188.
[4]	C. W. Curtis and I. Reiner, “Methods of Representation Theory,” Wiley, New York, 1981, pp. 177-184.
[5]	J. E. Roseblade, “A Note on Subnormal Coalition Cla- sses,” Mathematische Zeitschrift, Vol. 90, No. 5, 1965, pp. 373-375. doi:10.1007/BF01112356
[6]	C. F. Miller III, “On Group Theoretic Decision Problems and Their Classification,” Annals of Mathematics Studies, No. 68, Princeton University Press, Princeton, 1971, pp. 112-134.
[7]	M. F. Newman, “The Soluble Length of Soluble Linear Groups,” Mathematische Zeitschrift, Vol. 126, No.1, 1972, pp. 59-70. doi:10.1007/BF01580356
[8]	B. Li and A. Skiba, “New Characterizations of Finite Supersoluble Groups,” Science in China Series A: Mathe- matics, Vol. 51, No. 5, 2008, pp. 827- 841.
[9]	W. Guo and A. Skiba, “X-Permutable Maximal Subgroups of Sylow Subgroups of Finite Groups,” Ukrainian Mathematical Journal, Vol. 58, No. 10, 2006, pp. 1299- 1309.
[10]	W. Guo, K. Shum and A. Skiba, “X-Semipermutable Subgroups of Finite Groups,” Journal of Algebra, Vol. 315, No. 1, 2007, pp. 31-41. doi:10.1016/j.jalgebra.2007.06.002