Monatshefte für Mathematik

, Volume 171, Issue 2, pp 205–216 | Cite as

Finite groups with some NR-subgroups or \({\mathcal{H}}\)-subgroups

  • Izabela Agata MalinowskaEmail author
Open Access


Berkovich investigated the following concept: a subgroup H of a finite group G is called an NR-subgroup (Normal Restriction) if whenever \({K \trianglelefteq H}\), then \({K^G \cap H = K}\), where K G is the normal closure of K in G. Bianchi, Gillio Berta Mauri, Herzog and Verardi proved a characterization of soluble T-groups by means of \({\mathcal{H}}\)-subgroups: a subgroup H of G is said to be an \({\mathcal{H}}\)-subgroup of G if \({H^g \cap N_G(H) \leq H}\) for all \({g \in G}\). In this article we give new characterizations of finite soluble PST-groups in terms of NR-subgroups or \({\mathcal{H}}\)-subgroups. We will show that they are different from the ones given by Ballester-Bolinches, Esteban-Romero and Pedraza-Aguilera. Robinson established the structure of minimal non-PST-groups. We give the classification of groups all of whose second maximal subgroups (of even order) are soluble PST-groups.


NR-subgroups \({\mathcal{H}}\)-subgroups T-groups PT-groups PST-groups 

Mathematics Subject Classification (2000)

20D10 20D20 


Open Access

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Copyright information

© The Author(s) 2012

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of BiałystokBiałystokPoland

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