Home On the ๐œŽ-nilpotent hypercenter of finite groups
Article
Licensed
Unlicensed Requires Authentication

On the ๐œŽ-nilpotent hypercenter of finite groups

  • Viachaslau I. Murashka ORCID logo EMAIL logo and Alexander F. Vasilโ€™ev
Published/Copyright: May 7, 2022

Abstract

Let ๐œŽ be a partition of the set of all primes, and let ๐”‰ denote a hereditary formation. We describe all formations ๐”‰ for which the ๐”‰-hypercenter and the intersection of weak ๐พ-๐”‰-subnormalizers of all Sylow subgroups coincide in every finite group. In particular, the formation of all ๐œŽ-nilpotent groups has this property. With the help of our results, we solve a particular case of Shemetkovโ€™s problem about the intersection of ๐”‰-maximal subgroups and the ๐”‰-hypercenter. As a corollary, we obtain Hallโ€™s classical result about the hypercenter. We prove that the non-๐œŽ-nilpotent graph of a group is connected and its diameter is at most 3.

Acknowledgements

We would like to thank the reviewer for carefully reading our manuscript.

  1. Communicated by: Evgenii I. Khukhro

References

[1] A. Abdollahi and M. Zarrin, Non-nilpotent graph of a group, Comm. Algebra 38 (2010), no. 12, 4390โ€“4403. 10.1080/00927870903386460Search in Google Scholar

[2] S. Aivazidis, I.โ€‰N. Safonova and A.โ€‰N. Skiba, Subnormality and residuals for saturated formations: A generalization of Schenkmanโ€™s theorem, J. Group Theory 24 (2021), no. 4, 807โ€“818. 10.1515/jgth-2020-0149Search in Google Scholar

[3] R. Baer, Group elements of prime power index, Trans. Amer. Math. Soc. 75 (1953), 20โ€“47. 10.1090/S0002-9947-1953-0055340-0Search in Google Scholar

[4] R. Baer, Supersoluble immersion, Canad. J. Math. 11 (1959), 353โ€“369. 10.4153/CJM-1959-036-2Search in Google Scholar

[5] A. Ballester-Bolinches and L.โ€‰M. Ezquerro, Classes of Finite Groups, Math. Appl. (Springer) 584, Springer, Dordrecht, 2006. Search in Google Scholar

[6] A. Ballester-Bolinches, L.โ€‰M. Ezquerro and A.โ€‰N. Skiba, On subgroups of hypercentral type of finite groups, Israel J. Math. 199 (2014), no. 1, 259โ€“265. 10.1007/s11856-013-0030-ySearch in Google Scholar

[7] A. Ballester-Bolinches, S.โ€‰F. Kamornikov, M.โ€‰C. Pedraza-Aguilera and V. Pรฉrez-Calabuig, On ๐œŽ-subnormality criteria in finite ๐œŽ-soluble groups, Rev. R. Acad. Cienc. Exactas Fรญs. Nat. Ser. A Mat. RACSAM 114 (2020), no. 2, Paper No. 94. 10.1007/s13398-020-00824-4Search in Google Scholar

[8] A. Ballester-Bolinches and M.โ€‰D. Pรฉrez-Ramos, On ๐”‰-critical groups, J. Algebra 174 (1995), no. 3, 948โ€“958. 10.1006/jabr.1995.1161Search in Google Scholar

[9] A. Ballester-Bolinches and M.โ€‰D. Pรฉrez-Ramos, On a question of L.โ€‰A. Shemetkov, Comm. Algebra 27 (1999), no. 11, 5615โ€“5618. 10.1080/00927879908826777Search in Google Scholar

[10] J.โ€‰C. Beidleman and H. Heineken, A note on intersections of maximal โ„ฑ-subgroups, J. Algebra 333 (2011), 120โ€“127. 10.1016/j.jalgebra.2010.10.017Search in Google Scholar

[11] C. Cao, W. Guo and C. Zhang, On the structure of N ฯƒ -critical groups, Monatsh. Math. 189 (2019), no. 2, 239โ€“242. 10.1007/s00605-018-1201-zSearch in Google Scholar

[12] R. W. Carter, Nilpotent self-normalizing subgroups and system normalizers, Proc. Lond. Math. Soc. (3) 12 (1962), 535โ€“563. 10.1112/plms/s3-12.1.535Search in Google Scholar

[13] K. Doerk and T. Hawkes, Finite Soluble Groups, De Gruyter Exp. Math. 4, De Gruyter, Berlin, 1992. 10.1515/9783110870138Search in Google Scholar

[14] C.โ€‰J. Graddon, The relation between ๐”‰-reducers and ๐”‰-subnormalizers in finite soluble groups, J. London Math. Soc. (2) 4 (1971), 51โ€“61. 10.1112/jlms/s2-4.1.51Search in Google Scholar

[15] R.โ€‰L. Griess and P. Schmid, The Frattini module, Arch. Math. (Basel) 30 (1978), no. 3, 256โ€“266. 10.1007/BF01226050Search in Google Scholar

[16] W. Guo, Structure Theory for Canonical Classes of Finite Groups, Springer, Heidelberg, 2015. 10.1007/978-3-662-45747-4Search in Google Scholar

[17] P. Hall, On the system normalizers of a soluble group, Proc. Lond. Math. Soc. (2) 43 (1937), no. 7, 507โ€“528. 10.1112/plms/s2-43.6.507Search in Google Scholar

[18] B. Hu, J. Huang and A.โ€‰N. Skiba, Characterizations of finite ๐œŽ-nilpotent and ๐œŽ-quasinilpotent groups, Bull. Malays. Math. Sci. Soc. 42 (2019), no. 5, 2091โ€“2104. 10.1007/s40840-017-0593-6Search in Google Scholar

[19] B. Huppert, Zur Theorie der Formationen, Arch. Math. (Basel) 19 (1969), 561โ€“574. 10.1007/BF01899382Search in Google Scholar

[20] L.โ€‰S. Kazarin, A. Martรญnez-Pastor and M.โ€‰D. Pรฉrez-Ramos, On the Sylow graph of a group and Sylow normalizers, Israel J. Math. 186 (2011), 251โ€“271. 10.1007/s11856-011-0138-xSearch in Google Scholar

[21] O.โ€‰H. Kegel, Untergruppenverbรคnde endlicher Gruppen, die den Subnormalteilerverband echt enthalten, Arch. Math. (Basel) 30 (1978), no. 3, 225โ€“228. 10.1007/BF01226043Search in Google Scholar

[22] A.-M. Liu, W. Guo, I.โ€‰N. Safonova and A.โ€‰N. Skiba, ๐บ-covering subgroup systems for some classes of ๐œŽ-soluble groups, J. Algebra 585 (2021), 280โ€“293. 10.1016/j.jalgebra.2021.06.010Search in Google Scholar

[23] A. Lucchini and D. Nemmi, The diameter of the non-nilpotent graph of a finite group, Trans. Comb. 9 (2020), no. 2, 111โ€“114. Search in Google Scholar

[24] A. Mann, System normalizers and subnormalizers, Proc. Lond. Math. Soc. (3) 20 (1970), 123โ€“143. 10.1112/plms/s3-20.1.123Search in Google Scholar

[25] V.โ€‰I. Murashka, One one generalization of Baerโ€™s theorems about hypercenter and nilpotent residual, Probl. Fiz. Mat. Tekh. 16 (2013), 84โ€“88. Search in Google Scholar

[26] V.โ€‰I. Murashka, Classes of finite groups with generally subnormal cyclic primary subgroups, Sib. Math. J. 55 (2014), no. 6, 1353โ€“1367. 10.1134/S0037446614060135Search in Google Scholar

[27] V.โ€‰I. Murashka, Finite groups with given sets of ๐”‰-subnormal subgroups, Asian-Eur. J. Math. 13 (2020), no. 4, Article ID 2050073. 10.1142/S1793557120500734Search in Google Scholar

[28] D. Nemmi, Graphs encoding properties of finite groups, Masterโ€™s thesis, Universitร  degli Studi di Padova, 2020. Search in Google Scholar

[29] P. Schmid, The hypercenter of a profinite group, Beitr. Algebra Geom. 55 (2014), no. 2, 645โ€“648. 10.1007/s13366-013-0149-3Search in Google Scholar

[30] L.โ€‰A. Shemetkov, Graduated formations of groups, Math. USSR-Sb. 23 (1974), no. 4, 593โ€“611. 10.1070/SM1974v023n04ABEH002184Search in Google Scholar

[31] L.โ€‰A. Shemetkov, Factorizaton of nonsimple finite groups (in Russian), Algebra i Logika 15 (1976), no. 6, 684โ€“715, 744. 10.1007/BF01877483Search in Google Scholar

[32] L.โ€‰A. Shemetkov, Frattini extensions of finite groups and formations, Comm. Algebra 25 (1997), no. 3, 955โ€“964. 10.1080/00927879708825900Search in Google Scholar

[33] L.โ€‰A. Shemetkov and A.โ€‰N. Skiba, Formations of Algebraic Systems (in Russian), โ€œNaukaโ€, Moscow, 1989. Search in Google Scholar

[34] A.โ€‰N. Skiba, On the โ„ฑ-hypercentre and the intersection of all โ„ฑ-maximal subgroups of a finite group, J. Pure Appl. Algebra 216 (2012), no. 4, 789โ€“799. 10.1016/j.jpaa.2011.10.006Search in Google Scholar

[35] A.โ€‰N. Skiba, On ๐œŽ-subnormal and ๐œŽ-permutable subgroups of finite groups, J. Algebra 436 (2015), 1โ€“16. 10.1016/j.jalgebra.2015.04.010Search in Google Scholar

[36] A.โ€‰N. Skiba, Some characterizations of finite ๐œŽ-soluble P โข ฯƒ โข T -groups, J. Algebra 495 (2018), 114โ€“129. 10.1016/j.jalgebra.2017.11.009Search in Google Scholar

[37] A.โ€‰F. Vasilโ€™ev, T.โ€‰I. Vasilโ€™eva and A.โ€‰S. Vegera, Finite groups with a generalized subnormal embedding of Sylow subgroups, Sib. Math. J. 57 (2016), no. 2, 200โ€“212. 10.1134/S0037446616020038Search in Google Scholar

[38] A.โ€‰F. Vasilyev and V.โ€‰I. Murashka, Arithmetic graphs and classes of finite groups, Sib. Mat. J. 60 (2019), no. 1, 41โ€“55. 10.1134/S0037446619010051Search in Google Scholar

[39] X. Yi and S.โ€‰F. Kamornikov, Subgroup-closed lattice formations, J. Algebra 444 (2015), 143โ€“151. 10.1016/j.jalgebra.2015.07.021Search in Google Scholar

Received: 2021-08-30
Revised: 2022-01-31
Published Online: 2022-05-07
Published in Print: 2022-11-01

ยฉ 2022 Walter de Gruyter GmbH, Berlin/Boston

Downloaded on 20.9.2025 from https://www.degruyterbrill.com/document/doi/10.1515/jgth-2021-0138/html
Scroll to top button