A characterization of soluble groups in which normality is a transitive relation

Document Type : Research Paper

Author

University of Salerno

Abstract

A subgroup $X$ of‎ ‎a group $G$ is said to be an H-‎subgroup if‎ ‎NG(X) ∩ Xg  ≤ X for each element $g$ belonging to $G$‎. ‎In [M‎. ‎Bianchi and e.a.‎, ‎On finite soluble groups in which normality is a transitive relation‎, J‎. ‎Group Theory, ‎ 3 (2000) 147--156.] the authors showed that finite groups in which every subgroup has the H‎-‎property are exactly soluble groups in which normality is a transitive relation‎. ‎Here we extend this characterization to groups without simple sections‎.

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References

[1] A. Ballester-Bolinches and R. Esteban-Romero, On finite T-groups, J. Austral. Math. Soc. Ser. A, 75 (2003) 1–11.
[2] M. Bianchi, A. Gillio Berta Mauri, M. Herzog and L. Verardi, On finite solvable groups in which normality is a transitive relation, J. Group Theory, 3 (2000) 147–156.
[3] R. Esteban-Romero and G. Vincenzi, On generalized FC-groups in which normality is a transitive relation, J.Austral. Math. Soc. (2), 100 (2016) 192–198.
[4] W. Gaschütz, Gruppen, in denen das Normalteilersein transitiv ist, J. Reine Angew. Math., 198 (1957) 87–92.
[5] F. de Giovanni and G. Vincenzi, Pronormality in infinite groups, Math. Proc. R. Ir. Acad. Ser. A, 100 (2000)189–203.
[6] F. de Giovanni and G. Vincenzi, Some topics in the theory of pronormal subgroups of groups, Topics in infinite groups, 8, Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta, 2001 175–202.
[7] F. de Giovanni and G. Vincenzi, Pseudonormal subgroups of groups, Ricerche Mat., 52 (2003) 91–101.
[8] G. Kaplan, On finite T-groups and the Wielandt subgroup, J. Group Theory, 14 (2011) 855–863.
[9] G. Kaplan, On T-groups, supersolvable groups and maximal subgroups, Arch. Math. (Basel), 96 (2011) 19–25.
[10] G. Kaplan and G. Vincenzi, On the Wielandt subgroup of generalized FC-groups, Internat. J. Algebra Comput., 24 (2014) 1031–1042.
[11] E. I. Khukhro and V. D. Mazurov, The Kourovka Notebook, 16, Novosibirsk, Russian Academy of Sciences, 2006.
[12] L. G. Kovács, B. H. Neumann and H. de Vries, Some Sylow subgroups, Proc. Roy. Soc. Ser. A, 260 (1961) 304–316.
[13] N. F. Kuzennyi and I. Y. Subbotin, Groups with pronormal primary subgroups, Ukrainian. Math. J., 41 (1989) 286–289.
[14] T. A. Peng, Finite groups with pro-normal subgroups, Proc. Amer. Math. Soc., 20 (1969) 232–234.
[15] D. J. S. Robinson, Groups in which normality is a transitive relation, Proc. Cambridge Philos. Soc., 60 (1964) 21–38.
[16] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Springer, Berlin, 1972.
[17] D. J. S. Robinson, A. Russo and G. Vincenzi, On groups which contain no HNN-extensions, Internat. J. Algebra Comput., 17 (2007) 1377–1387.
[18] E. Romano and G. Vincenzi, Pronormality in generalized FC-groups, Bull. Aust. Math. Soc., 83 (2011) 220–230.
[19] E. Romano and G. Vincenzi, Groups in which normality is a weakly transitive relation, J. Algebra Appl., 14 (2015) 12 pp.
[20] J. S. Rose, Finite soluble groups with pronormal system normalizers, Proc. London Math. Soc. (3), 17 (1967) 447–469.
[21] A. Russo, On Groups in which normality is a transitive relation, Comm. Algebra, 40 (2012) 3950–3954.
[22] G. Vincenzi, Groups satisfying the maximal condition on non-pronormal subgroups, Algebra Colloq., 2 (1998) 121–134.
[23] G. Zacher, Caratterizzazione dei t-gruppi finiti risolubili, Ricerche Mat., 1 (1952) 287–294.

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History

  • Receive Date: 09 July 2015
  • Revise Date: 21 September 2015
  • Accept Date: 21 September 2015
  • Published Online: 01 March 2017

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