On perfectly embedded subgroups

Articles in Press

Document Type : Research Paper

Authors

Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, 84084, Fisciano (SA), Italy

Abstract

Let \(G\) be a group, and let \(H \le K \le G\) be subgroups. We say that \(H\) is \emph{perfectly embedded} in \(K\) if \([H,K]=H\). In the special case \(K=G\), we simply say that \(H\) is a perfectly embedded subgroup of \(G\).
 
The aim of this paper is to investigate several aspects of this notion. We first characterize the groups whose only perfectly embedded subgroup is the trivial subgroup, showing that they are exactly the hypocentral groups. At the other extreme, we study groups in which every normal subgroup is perfectly embedded, providing a complete classification of finite \(\mathcal T\)-groups with this feature.
 
We also examine groups in which every subgroup is perfectly embedded in an appropriate subgroup normalizing it. Finally, attention is restricted to abelian normal subgroups, focusing once more on the two opposite situations. In particular, we show that every finite Frobenius group belongs to the class of groups in which every abelian normal subgroup is perfectly embedded.

Keywords

Main Subjects

References

[1] M. Aschbacher, Finite Group Theory, Cambridge Stud. Adv. Math., 10 Cambridge Univ. Press, Cambridge, 1986.
[2] R. Chamberlain, Groups for which all normal subgroups are perfect, MathOverflow/Math StackExchange, (2018), available at https://math.stackexchange.com/questions/2804510.
[3] 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.
[4] C. Delizia, M. Gaeta and C. Monetta, On generalized concise words, J. Group Theory, 28 no. 2 (2025) 475–487.
[5] 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.
[6] C. Delizia, U. Jezernik, P. Moravec and C. Nicotera, Groups in which every non-cyclic subgroup contains its centralizer, J. Algebra Appl., 13 no. 5 (2014) 1350154 (11 pages).
[7] C. Delizia, U. Jezernik, P. Moravec and C. Nicotera, Groups in which every non-nilpotent subgroup is self-normalizing, Ars Math. Contemp., 15 no. 1 (2018) 39–51.
[8] 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 no. 2 (2017) 401–410.
[9] C. Delizia and C. Nicotera, Groups with many self-centralizing or self-normalizing subgroups, Int. J. Group Theory, 9 no. 1 (2020) 43–57.
[10] C. Delizia, P. Shumyatsky and A. Tortora, On groups with finite conjugacy classes in a verbal subgroup, Bull. Aust. Math. Soc., 96 (2017) 429–437.
[11] C. Delizia, P. Shumyatsky and A. Tortora, On semiconcise words, J. Group Theory, 23 (2020) 629–639.
[12] C. Delizia, P. Shumyatsky, A. Tortora and M. Tota, On conciseness of some commutator words, Arch. Math., 112 no. 1 (2019) 27–32.
[13] The GAP Group, GAP – Groups, Algorithms, and Programming, Version 4.11.1, 2021, available at https://www.gap-system.org.
[14] D. J. S. Robinson, A Course in the Theory of Groups, Grad. Texts Math., 80 Springer, New York, 2nd edn., 1996.
[15] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vol. 1, Springer, Berlin, 1972.
[16] D. J. S. Robinson, Finiteness Conditions and Generalized Soluble Groups, Vol. 2, Springer, Berlin, 1973.
International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 18 July 2026

Files

History

  • Receive Date: 23 May 2026
  • Revise Date: 23 June 2026
  • Accept Date: 03 July 2026
  • Published Online: 18 July 2026

Share

How to cite

Statistics