Groups whose subgroups satisfy the weak subnormalizer condition

  • R. Esteban Romero
  • F. de GiovanniEmail author
  • A. Russo
Original Paper


A subgroup X of a group G is said to satisfy the weak subnormalizer condition if \(N_G(Y)\le N_G(X)\) for each non-normal subgroup Y of G such that \(X\le Y\le N_G(X)\). The behaviour of generalized soluble groups whose (cyclic) subgroups satisfy the weak subnormalizer condition is investigated.


Weak subnormalizer condition T-group Weakly radical group 

Mathematics Subject Classification

20E15 20F16 



  1. Ballester Bolinches, A., Esteban Romero, R., Asaad, M.: Products of Finite Groups. de Gruyter, Berlin (2010)CrossRefzbMATHGoogle Scholar
  2. de Giovanni, F., Vincenzi, G.: Pseudonormal subgroups of groups. Ricerche Mat. 52, 91–101 (2003)MathSciNetzbMATHGoogle Scholar
  3. Esteban Romero, R., Vincenzi, G.: Some characterizations of groups in which normality is a transitive relation by means of subgroup embedding properties. Int. J. Group Theory 7(2), 9–16 (2018)MathSciNetGoogle Scholar
  4. Gaschütz, W.: Gruppen in denen das Normalteilersein transitiv ist. J. Reine Angew. Math. 198, 87–92 (1957)MathSciNetzbMATHGoogle Scholar
  5. Heineken, H.: A class of three-Engel groups. J. Algebra 17, 341–345 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  6. Kurdachenko, L.A., Otal, J.: On the influence of transitively normal subgroups on the structure of some infinite groups. Publ. Mat. 57, 83–106 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  7. Mahdavianary, S.K.: A special class of three-Engel groups. Arch. Math. (Basel) 40, 193–199 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. Mysovskikh, V.I.: Subnormalizers and embedding properties of subgroups of finite groups. J. Math. Sci. (New York) 112, 4386–4397 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. Robinson, D.J.S.: Groups in which normality is a transitive relation. Proc. Camb. Philos. Soc. 60, 21–38 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  10. Robinson, D.J.S.: A note on finite groups in which normality is transitive. Proc. Am. Math. Soc. 19, 933–937 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  11. Robinson, D.J.S.: Finiteness Conditions and Generalized Soluble Groups. Springer, Berlin (1972)CrossRefzbMATHGoogle Scholar
  12. Robinson, D.J.S.: Finite groups whose cyclic subnormal subgroups are permutable. Algebra Colloq. 12, 171–180 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  13. Sakamoto, T.: Groups whose cyclic subnormal subgroups are normal. Bull. Kyushu Inst. Tech. Pure Appl. Math. 48, 15–19 (2001)MathSciNetzbMATHGoogle Scholar
  14. Xu, M.: Solubility of \(AT\)-groups and infinite \(IT\)-groups. Acta Math. Sin. 31, 663–670 (1988)zbMATHGoogle Scholar

Copyright information

© The Managing Editors 2019

Authors and Affiliations

  1. 1.Institut Universitari de Matemàtiques Pura i AplicadaUniversitat Politècnica de ValènciaValenciaSpain
  2. 2.Departament de MatemàtiquesUniversitat de ValènciaValenciaSpain
  3. 3.Dipartimento di Matematica e ApplicazioniUniversità di Napoli Federico IINaplesItaly
  4. 4.Dipartimento di Matematica e FisicaUniversità della CampaniaCasertaItaly

Personalised recommendations