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FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS
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 Title & Authors
FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS
Shen, Zhencai; Shi, Wujie; Zhang, Jinshan;
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 Abstract
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 to be intersection of the normalizers of all non-cyclic subgroups of G. Set . Define for . By denote the terminal term of the ascending series. It is proved that if , then the derived subgroup G' is nilpotent. Furthermore, if all elements of prime order or order 4 of G are in , then G' is also nilpotent.
 Keywords
derived subgroup;meta-nilpotent group;solvable group;nilpotency class;fitting length;
 Language
English
 Cited by
1.
On the generalized norm of a finite group, Journal of Algebra and Its Applications, 2016, 15, 01, 1650008  crossref(new windwow)
2.
Groups with Certain Normality Conditions, Communications in Algebra, 2016, 44, 8, 3308  crossref(new windwow)
3.
On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups, Communications in Algebra, 2013, 41, 5, 1608  crossref(new windwow)
4.
Generalised norms in finite soluble groups, Journal of Algebra, 2014, 402, 392  crossref(new windwow)
 References
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. crossref(new window)

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. crossref(new window)

9.
R. Brandl, Groups with few non-normal subgroups, Comm. Algebra 23 (1995), no. 6, 2091-2098. crossref(new window)

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. crossref(new window)

11.
J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1970), no. 1, 15-17. crossref(new window)

12.
A. R. Camina, The Wielandt length of finite groups, J. Algebra 15 (1970), 142-148. crossref(new window)

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. crossref(new window)

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. crossref(new window)

20.
H. Mousavi, On Finite groups with few non-normal subgroups, Comm. Algebra 27 (1999), no. 7, 3143-3151. crossref(new window)

21.
M. Newman and J. Wiegold, Groups with many nilpotent subgroups, Arch. Math. 15 (1964), 241-250. crossref(new window)

22.
D. S. Passman, Nonnormal subgroups of p-groups, J. Algebra 15 (1970), 352-370. crossref(new window)

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. crossref(new window)

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. crossref(new window)