FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS

Title & Authors
FINITE NON-NILPOTENT GENERALIZATIONS OF HAMILTONIAN GROUPS
Shen, Zhencai; Shi, Wujie; Zhang, Jinshan;

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 $\small{\mathcal{A}(G)}$ to be intersection of the normalizers of all non-cyclic subgroups of G. Set $\mathcal{A}_0 Keywords derived subgroup;meta-nilpotent group;solvable group;nilpotency class;fitting length; Language English Cited by 1. On the generalized norm of a finite group, Journal of Algebra and Its Applications, 2016, 15, 01, 1650008 2. Groups with Certain Normality Conditions, Communications in Algebra, 2016, 44, 8, 3308 3. On a Generalization of Hamiltonian Groups and a Dualization of PN-Groups, Communications in Algebra, 2013, 41, 5, 1608 4. Generalised norms in finite soluble groups, Journal of Algebra, 2014, 402, 392 References 1. R. Baer, Der Kern eine charakteristische Untergruppe, Compositio Math. 1 (1934), 254-283. 2. R. Baer, Norm and hypernorm, Publ. Math. Debrecen 4 (1956), 347-356. 3. R. Baer, Classes of finite groups and their properties, Illinois J. Math. 1 (1957), 115-187. 4. R. Baer, Zentrum und Kern von Gruppen mit Elementen unendlicher Ordnung, Compositio Math. 2 (1935), 247-249. 5. J. C. Beidleman, H. Heineken, and M. Newell, Centre and norm, Bull. Austral. Math. Soc. 69 (2004), no. 3, 457-464. 6. T. R. Berger, L. G. Kovacs, and M. F. Newman, Groups of prime power order with cyclic Frattini subgroup, Nederl. Akad.Wetensch. Indag. Math. 42 (1980), no. 1, 13-18. 7. Y. Berkovich and Z. Janko, Structure of finite p-groups with given subgroups, Ischia group theory 2004, 13-93, Contemp. Math., 402, Amer. Math. Soc., Providence, RI, 2006. 8. Z. Bozikov and Z. Janko, A complete classification of finite p-groups all of whose noncyclic subgroups are normal, Glas. Mat. Ser. III 44(64) (2009), no. 1, 177-185. 9. R. Brandl, Groups with few non-normal subgroups, Comm. Algebra 23 (1995), no. 6, 2091-2098. 10. R. A. Bryce and J. Cossey, The Wielandt subgroup of a finite soluble group, J. London Math. Soc. (2) 40 (1989), no. 2, 244-256. 11. J. Buckley, Finite groups whose minimal subgroups are normal, Math. Z. 116 (1970), no. 1, 15-17. 12. A. R. Camina, The Wielandt length of finite groups, J. Algebra 15 (1970), 142-148. 13. Z. Chen, Inner$\Sigma\$-groups. II, Acta Math. Sinica 24 (1981), no. 3, 331-336.

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