• Shen, Zhencai (LMAM and School of Mathematical Sciences Peking University) ;
  • Shi, Wujie (School of Mathematics and Statics Chongqing University of Arts and Sciences) ;
  • Zhang, Jinshan (School of Science Sichuan University of Science and Engineering)
  • Received : 2010.04.07
  • Published : 2011.11.30


In J. Korean Math. Soc, Zhang, Xu and other authors investigated the following problem: what is the structure of finite groups which have many normal subgroups? In this paper, we shall study this question in a more general way. For a finite group G, we define the subgroup $\mathcal{A}(G)$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0=1$. Define $\mathcal{A}_{i+1}(G)/\mathcal{A}_i(G)=\mathcal{A}(G/\mathcal{A}_i(G))$ for $i{\geq}1$. By $\mathcal{A}_{\infty}(G)$ denote the terminal term of the ascending series. It is proved that if $G=\mathcal{A}_{\infty}(G)$, then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in $\mathcal{A}(G)$, then G' is also nilpotent.


Supported by : Natural Science Foundation of China, NSF


  1. R. Baer, Der Kern eine charakteristische Untergruppe, Compositio Math. 1 (1934), 254-283.
  2. R. Baer, Norm and hypernorm, Publ. Math. Debrecen 4 (1956), 347-356.
  3. R. Baer, Classes of finite groups and their properties, Illinois J. Math. 1 (1957), 115-187.
  4. R. Baer, Zentrum und Kern von Gruppen mit Elementen unendlicher Ordnung, Compositio Math. 2 (1935), 247-249.
  5. J. C. Beidleman, H. Heineken, and M. Newell, Centre and norm, Bull. Austral. Math. Soc. 69 (2004), no. 3, 457-464.
  6. T. R. Berger, L. G. Kovacs, and M. F. Newman, Groups of prime power order with cyclic Frattini subgroup, Nederl. Akad.Wetensch. Indag. Math. 42 (1980), no. 1, 13-18.
  7. Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, Ischia group theory 2004, 13-93, Contemp. Math., 402, Amer. Math. Soc., Providence, RI, 2006.
  8. Z. Bozikov and Z. Janko, A complete classification of finite p-groups all of whose noncyclic subgroups are normal, Glas. Mat. Ser. III 44(64) (2009), no. 1, 177-185.
  9. R. Brandl, Groups with few non-normal subgroups, Comm. Algebra 23 (1995), no. 6, 2091-2098.
  10. R. A. Bryce and J. Cossey, The Wielandt subgroup of a finite soluble group, J. London Math. Soc. (2) 40 (1989), no. 2, 244-256.
  11. J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1970), no. 1, 15-17.
  12. A. R. Camina, The Wielandt length of finite groups, J. Algebra 15 (1970), 142-148.
  13. Z. Chen, Inner $\Sigma$-groups. II, Acta Math. Sinica 24 (1981), no. 3, 331-336.
  14. D. Gorenstein, Finite Groups, Chelsea, New York, 1980.
  15. B. Huppert, Endliche Gruppen I, Springer-Verlag, Berlin, Heidelberg, New York, 1967.
  16. H. Kurzweil and B. Stellmacher, The Theory of Finite Groups: An Introduction, Springer-Verlag New York, Inc., 2004.
  17. S. Li and Z. Shen, On the intersection of normalizers of direved subgroups of all subgroups of a finite group, J. Algebra 323 (2010), 1349-1357.
  18. T. D. Lukashova, On the noncyclic norm of infinite locally finite groups, Ukrain. Mat. Zh. 54 (2002), no. 3, 342-348; translation in Ukrainian Math. J. 54 (2002), no. 3, 421-428.
  19. G. A. Miller and H. C. Moreno, Non-abelian groups in which every subgroup is abelian, Trans. Amer. Math. Soc. 4 (1903), no. 4, 398-404.
  20. H. Mousavi, On Finite groups with few non-normal subgroups, Comm. Algebra 27 (1999), no. 7, 3143-3151.
  21. M. Newman and J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15 (1964), 241-250.
  22. D. S. Passman, Nonnormal subgroups of p-groups, J. Algebra 15 (1970), 352-370.
  23. D. J. S. Robinson, A Course in the Theory of Groups, Springer-Verlag, New York, 1982.
  24. E. Schenkman, On the norm of a group, Illinois J. Math. 4 (1960), 150-152.
  25. Q. Song and H. Qu, Finite 2-groups whose subgroups are cyclic or normal, Math. Pract. Theory 38 (2008), no. 10, 191-197.
  26. H. Wielandt, Uber der Normalisator der Subnormalen Untergruppen, Math. Z. 69 (1958), 463-465.
  27. Q. Zhang, X. Guo, H. Qu, and M. Xu, Finite Groups which have many normal subgroups, J. Korean Math. Soc. 46 (2009), no. 6, 1165-1178.

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