Abstract
A Schmidt group is a finite non-nilpotent group such that every proper subgroup is nilpotent.
In this paper, we prove that if every Schmidt subgroup of a finite group πΊ is subnormal or modular, then
-
Communicated by: Evgenii I. Khukhro
References
[1] K.βA. Al-Sharo and A.βN. Skiba, On finite groups with π-subnormal Schmidt subgroups, Comm. Algebra 45 (2017), no. 10, 4158β4165. 10.1080/00927872.2016.1236938Search in Google Scholar
[2] A. Ballester-Bolinches, R. Esteban-Romero and D.βJ.βS. Robinson, On finite minimal non-nilpotent groups, Proc. Amer. Math. Soc. 133 (2005), no. 12, 3455β3462. 10.1090/S0002-9939-05-07996-7Search in Google Scholar
[3] A. Ballester-Bolinches and L.βM. Ezquerro, Classes of Finite Groups, Math. Appl. 584, Springer, Dordrecht, 2006. Search in Google Scholar
[4] A. Ballester-Bolinches, S.βF. Kamornikov and X. Yi, Finite groups with π-subnormal Schmidt subgroups, Bull. Malays. Math. Sci. Soc. 45 (2022), no. 5, 2431β2440. 10.1007/s40840-022-01369-ySearch in Google Scholar
[5] I.βV. Bliznets and V.βM. Selkin, On finite groups with modular Schmidt subgroup, Probl. Fiz. Mat. Tekh. (2019), no. 4(41), 36β38. Search in Google Scholar
[6] W. GaschΓΌtz, Lectures on Subgroups of Sylow Type in Finite Soluble Groups, Notes Pure Math. 11, Australian National University, Canberra, 1979. Search in Google Scholar
[7] Y.βA. Golfand, On groups all of whose subgroups are special, Dokl. Akad. Nauk SSSR (N.βS.) 60 (1948),1313β1315. Search in Google Scholar
[8] W. Guo, I.βN. Safonova and A.βN. Skiba, On π-subnormal subgroups of finite groups, Southeast Asian Bull. Math. 45 (2021), no. 6, 813β824. Search in Google Scholar
[9] B. Hu and J. Huang, On finite groups with generalized π-subnormal Schmidt subgroups, Comm. Algebra 46 (2018), no. 7, 3127β3134. 10.1080/00927872.2017.1404091Search in Google Scholar
[10] B. Hu, J. Huang, D. Song and I.βN. Safonova, Finite groups with πΎ-π-subnormal Schmidt subgroups, Comm. Algebra 49 (2021), no. 10, 4513β4518. 10.1080/00927872.2021.1923023Search in Google Scholar
[11] J. Huang, B. Hu and A.βN. Skiba, Finite groups with weakly subnormal and partially subnormal subgroups, Sib. Math. J. 62 (2021), no. 1, 169β177. 10.1134/S0037446621010183Search in Google Scholar
[12] J. Huang, B. Hu and X. Zheng, Finite groups whose π-maximal subgroups are modular, Sib. Math. J. 59 (2018), no. 3, 556β564. 10.1134/S0037446618030187Search in Google Scholar
[13] B. Huppert, Endliche Gruppen. I, Grundlehren Math. Wiss. 134, Springer, Berlin, 1967. 10.1007/978-3-642-64981-3Search in Google Scholar
[14] I.βM. Isaacs, Finite Group Theory, Grad. Stud. Math. 92, American Mathematical Society, Providence, 2008. Search in Google Scholar
[15] V.βN. Knyagina and V.βS. Monakhov, On finite groups with some subnormal Schmidt subgroups, Sib. Math. J. 45 (2004), no. 6, 1075β1079. 10.1023/B:SIMJ.0000048922.59466.20Search in Google Scholar
[16] V.βS. Monakhov, Schmidt subgroups, their existence and some applications, Ukrainian Mathematics Congressβ2001, Inst. Mat. NAN Ukrainy, Kiev (2002), 81β90. Search in Google Scholar
[17] V.βS. Monakhov and I.βL. Sokhor, Finite groups with restrictions on normal subgroups, Publ. Math. Debrecen 89 (2016), no. 1β2, 243β252. 10.5486/PMD.2016.7541Search in Google Scholar
[18] O.βY. Schmidt, Groups all subgroups of which are special, Mat. Sb. 31 (1924), no. 3β4, 366β372. Search in Google Scholar
[19] R. Schmidt, Modulare Untergruppen endlicher Gruppen, Illinois J. Math. 13 (1969), 358β377. 10.1215/ijm/1334250798Search in Google Scholar
[20] R. Schmidt, Modular subgroups of finite groups. II, Illinois J. Math. 14 (1970), 344β362. 10.1215/ijm/1256053189Search in Google Scholar
[21] R. Schmidt, Subgroup Lattices of Groups, De Gruyter Exp. Math. 14, Walter de Gruyter, Berlin, 1994. 10.1515/9783110868647Search in Google Scholar
[22] A.βN. Skiba, On π-subnormal and π-permutable subgroups of finite groups, J. Algebra 436 (2015), 1β16. 10.1016/j.jalgebra.2015.04.010Search in Google Scholar
[23] M. Suzuki, On the lattice of subgroups of finite groups, Trans. Amer. Math. Soc. 70 (1951), 345β371. 10.1090/S0002-9947-1951-0039717-3Search in Google Scholar
[24] V.βA. Vedernikov, Finite groups with subnormal Schmidt subgroups, Algebra Logic 46 (2007), no. 6, 363β372. 10.1007/s10469-007-0036-9Search in Google Scholar
[25] X. Yi and S.βF. Kamornikov, Finite groups with π-subnormal Schmidt subgroups, J. Algebra 560 (2020), 181β191. 10.1016/j.jalgebra.2020.05.021Search in Google Scholar
Β© 2024 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- The relational complexity of linear groups acting on subspaces
- Cliques in derangement graphs for innately transitive groups
- Representation zeta function of a family of maximal class groups: Non-exceptional primes
- Character degrees of 5-groups of maximal class
- Automorphic word maps and the AmitβAshurst conjecture
- Groups with subnormal or modular Schmidt ππ-subgroups
- Finite normal subgroups of strongly verbally closed groups
- The central tree property and algorithmic problems on subgroups of free groups
- Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
- Isomorphisms and commensurability of surface Houghton groups
Articles in the same Issue
- Frontmatter
- The relational complexity of linear groups acting on subspaces
- Cliques in derangement graphs for innately transitive groups
- Representation zeta function of a family of maximal class groups: Non-exceptional primes
- Character degrees of 5-groups of maximal class
- Automorphic word maps and the AmitβAshurst conjecture
- Groups with subnormal or modular Schmidt ππ-subgroups
- Finite normal subgroups of strongly verbally closed groups
- The central tree property and algorithmic problems on subgroups of free groups
- Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
- Isomorphisms and commensurability of surface Houghton groups