On some groups whose subnormal subgroups are contranormal-free

Kurdachenko, Leonid A.; Longobardi, Patrizia; Maj, Mercede

doi:10.22108/ijgt.2024.139136.1871

تعداد نشریات	43
تعداد شماره‌ها	1,724
تعداد مقالات	14,114
تعداد مشاهده مقاله	34,453,704
تعداد دریافت فایل اصل مقاله	13,795,555

	On some groups whose subnormal subgroups are contranormal-free
International Journal of Group Theory
مقاله 6، دوره 14، شماره 2، شهریور 2025، صفحه 99-115 اصل مقاله (467.46 K)
نوع مقاله: Ischia Group Theory 2022
شناسه دیجیتال (DOI): 10.22108/ijgt.2024.139136.1871
نویسندگان
Leonid A. Kurdachenko¹؛ Patrizia Longobardi²؛ Mercede Maj^* ²
¹Department of Algebra and Geometry, School of Mathematics and Mechanics, National Dnipro University, Gagarin Prospect 72, Dnipro 10, 49010 Ukraine
²Department of Mathematics, Università di Salerno, via Giovanni Paolo II, 132, 84084 Fisciano (Salerno), Italy
چکیده
If $G$ is a group, a subgroup $H$ of $G$ is said to be contranormal in $G$ if $H^G = G$, where $H^G$ is the normal closure of $H$ in $G$. We say that a group is contranormal-free if it does not contain proper contranormal subgroups. Obviously, a nilpotent group is contranormal-free. Conversely, if $G$ is a finite contranormal-free group, then $G$ is nilpotent. We study (infinite) groups whose subnormal subgroups are contranormal-free. We prove that if $G$ is a group which contains a normal nilpotent subgroup $A$ such that $G/A$ is a periodic Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is generated by subnormal nilpotent subgroups; in particular $G$ is a Baer group. Furthermore, if $G$ is a group which contains a normal nilpotent subgroup $A$ such that the $0$-rank of $A$ is finite, the set $\Pi(A)$ is finite, $G/A$ is a Baer group, and every subnormal subgroup of $G$ is contranormal-free, then $G$ is a Baer group.
کلیدواژه‌ها
Contranormal subgroups؛ subnormal subgroups؛ nilpotent groups؛ hypercentral groups؛ upper central series

مراجع
[1] M. R. Dixon, L. A. Kurdachenko and I. Ya. Subbotin, On the structure of some contranormal-free groups, Comm. Algebra, 49 no. 11 (2021) 4940–4946. [2] M. R. Dixon, L. A. Kurdachenko and I. Ya. Subbotin, On conormal subgroups, Internat. J. Algebra Comput., 2 no. 2 (2022) 327–345. [3] P. Hall, Some sufficient conditions for a group to be nilpotent, Illinois J. Math., 2 (1958) 787–801. [4] P. Hall, On the finiteness of certain soluble groups, Proc. London Math. Soc., 9 (1959) 595–622. [5] L. A. Kurdachenko, F C-groups with bounded orders of elements of the periodic part, Sib. Math. J., 6 (1975) 923–929. [6] L. A. Kurdachenko, P. Longobardi and M. Maj, On the structure of some locally nilpotent groups without contranormal subgroups, J. Group Theory, 25 (2022) 75–90. [7] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, On some criteria of nilpotency, Comm. Algebra, 30 no. 8 (2002) 3755–3776. [8] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, Groups with prescribed quotient groups and associated module theory, WORLD SCENTIFIC, New Jersey, 2002. [9] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, Artinian modules over group rings, Frontiers in Mathematics, BIRKHÄUSER, Basel, 2007. [10] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, Abnormal, pronormal, contranormal and Carter subgroups in some generalized minimax groups, Comm. Algebra, 33 no. 12 (2005) 4595–4616. [11] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, Criteria of nilpotency and influence of contranormal subgroups on the structure of infinite groups, Turkish J. Math., 33 (2009) 227–237. [12] L. A. Kurdachenko, J. Otal and I. Ya. Subbotin, On influence of contranormal subgroups on the structure of infinite groups, Comm. Algebra, 37 (2010) 4542–4557. [13] L. A. Kurdachenko, N. N. Semko and I. Ya. Subbotin, Insight into modules over Dedekind domains, Institute of Mathematics: Kyiv, 2008. [14] D. J. S. Robinson, A theorem of finitely generated hyperabelian groups, Invent. Math., 10 (1970) 38–43. [15] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, part 1, 2, SPRINGER, Berlin, 1972. [16] J. S. Rose, Nilpotent subgroups of finite soluble groups, Math. Z., 106 (1968) 97–112. [17] B. A. F. Wehrfritz, Groups with no proper contranormal subgroups, Publ. Mat., 64 (2020) 183–194. [18] D. I. Zaitsev, L. A. Kurdachenko and A. V. Tushev, The modules over nilpotent groups of finite rank, Algebra Logic, 24 no. 6 (1985) 412–436.
آمار تعداد مشاهده مقاله: 178 تعداد دریافت فایل اصل مقاله: 225

سامانه مدیریت نشریات علمی. طراحی و پیاده سازی از سیناوب

پیوندهای مفید

اخبار و اعلانات

آمار

On some groups whose subnormal subgroups are contranormal-free