Document Type : Ischia Group Theory 2018

**Authors**

Department of Mathematics, University of Salerno, Italy

**Abstract**

The purpose of this paper is to present a comprehensive overview of known and new results concerning the structure of groups in which all subgroups, except those having a given property, are either self-centralizing or self-normalizing.

**Keywords**

**Main Subjects**

[1] Y. Berkovich, Groups of prime power order, 1, Walter de Gruyter GmbH & Co. KG, Berlin, 2008.

[2] C. Casolo, Groups with all subgroups subnormal, Note Mat., 28 (2008) 1–149.

[3] M. De Falco and C. Musella, A normalizer condition for modular subgroups, Advances in Group Theory, Aracne,

Roma, 2002 163–172.

Roma, 2002 163–172.

[4] M. De Falco, F. de Giovanni and C. Musella, Groups with many self-normalizing subgroups, Algebra Discrete Math.,

4 (2009) 55–65.

4 (2009) 55–65.

[5] C. Delizia, U. Jezernik, P. Moravec and C. Nicotera, Groups in which every non-cyclic subgroup contains its

centralizer, J. Algebra Appl., 13 (2014) pp. 11.

centralizer, J. Algebra Appl., 13 (2014) pp. 11.

[6] C. Delizia, H. Dietrich, P. Moravec and C. Nicotera, Groups in which every non-abelian subgroup is self-centralizing,

J. Algebra, 462 (2016) 23–36.

J. Algebra, 462 (2016) 23–36.

[7] C. Delizia, U. Jezernik, P. Moravec, C. Nicotera and C. Parker, Locally finite groups in which every non-cyclic

subgroup is self-centralizing, J. Pure Appl. Algebra, 221 (2017) 401–410.

subgroup is self-centralizing, J. Pure Appl. Algebra, 221 (2017) 401–410.

[8] C. Delizia, U. Jezernik, P. Moravec and C. Nicotera, Groups in which every non-abelian subgroup is self-normalizing,

Monatsh. Math., 185 (2018) 591–600.

Monatsh. Math., 185 (2018) 591–600.

[9] C. Delizia, U. Jezernik, P. Moravec and C. Nicotera, Groups in which every non-nilpotent subgroup is selfnormalizing, Ars Math. Contemp., 15 (2018) 39–51.

[10] G. Giordano, Gruppi con normalizzatori estremali, Matematiche (Catania), 26 291–296 (1972).

[11] M. Hassanzadeh and Z. Mostaghim, Groups with few self-centralizing subgroups which are not self-normalizing,

Rend. Sem. Mat. Univ. Padova, in press.

Rend. Sem. Mat. Univ. Padova, in press.

[12] L. A. Kurdachenko and H. Smith, Groups with all subgroups either subnormal or self-normalizing, J. Pure Appl.

Algebra, 196 (2005) 271–278.

Algebra, 196 (2005) 271–278.

[13] L. A. Kurdachenko, J. Otal, A. Russo and G. Vincenzi, Groups whose all subgroups are ascendant or self-normalizing,

Cent. Eur. J. Math., 9 (2011) 420–432.

Cent. Eur. J. Math., 9 (2011) 420–432.

[14] D. J. S. Robinson, A course in the theory of groups, Springer-Verlag, New York, 2nd edition, 1996.

[15] R. Schmidt, Subgroup lattices of groups, de Gruyter Exp. Math., 14, Walter de Gruyter, Berlin, 2014.

[16] S. E. Stonehever, Permutable subgroups of infinite groups, Math. Z., 125 (1972) 1–16.

Proceedings of the Ischia Group Theory 2018

March 2020Pages 43-57

**Receive Date:**02 December 2018**Revise Date:**21 January 2019**Accept Date:**30 January 2019**Published Online:**01 March 2020