On perfectly embedded subgroups

Delizia, Costantino; Gaeta, Michele; Monetta, Carmine

doi:10.22108/ijgt.2026.148986.2023

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	On perfectly embedded subgroups
International Journal of Group Theory
مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 27 تیر 1405 اصل مقاله (424.04 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2026.148986.2023
نویسندگان
Costantino Delizia؛ Michele Gaeta؛ Carmine Monetta^*
Department of Mathematics, University of Salerno, Via Giovanni Paolo II 132, 84084, Fisciano (SA), Italy
چکیده
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.
کلیدواژه‌ها
Embedding of subgroups؛ commutator subgroups؛ perfect groups؛ hypocentral groups

مراجع
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On perfectly embedded subgroups