Hostname: page-component-669899f699-tpknm Total loading time: 0 Render date: 2025-05-05T14:57:43.803Z Has data issue: false hasContentIssue false
  • English
  • Français

A CHARACTERISATION OF SOLUBLE ${PST}$-GROUPS

Part of: Representation theory of groups

Published online by Cambridge University Press:  15 March 2024

ZHIGANG WANG Open the ORCID record for ZHIGANG WANG [Opens in a new window] ,
A-MING LIU Open the ORCID record for A-MING LIU [Opens in a new window] ,
VASILY G. SAFONOV Open the ORCID record for VASILY G. SAFONOV [Opens in a new window]  and
ALEXANDER N. SKIBA Open the ORCID record for ALEXANDER N. SKIBA [Opens in a new window]
ZHIGANG WANG
Affiliation:
School of Mathematics and Statistics, Hainan University, Haikou, Hainan 570228, PR China e-mail: [email protected]
A-MING LIU
Affiliation:
School of Mathematics and Statistics, Hainan University, Haikou, Hainan 570228, PR China e-mail: [email protected]
VASILY G. SAFONOV
Affiliation:
Institute of Mathematics of the National Academy of Sciences of Belarus, Minsk 220072, Belarus and Department of Mechanics and Mathematics, Belarusian State University, Minsk 220030, Belarus e-mail: [email protected], [email protected]
ALEXANDER N. SKIBA*
Affiliation:
Department of Mathematics and Technologies of Programming, Francisk Skorina Gomel State University, Gomel 246019, Belarus
*
e-mail: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Let G be a finite group. A subgroup A of G is said to be S-permutable in G if A permutes with every Sylow subgroup P of G, that is, $AP=PA$. Let $A_{sG}$ be the subgroup of A generated by all S-permutable subgroups of G contained in A and $A^{sG}$ be the intersection of all S-permutable subgroups of G containing A. We prove that if G is a soluble group, then S-permutability is a transitive relation in G if and only if the nilpotent residual $G^{\mathfrak {N}}$ of G avoids the pair $(A^{s G}, A_{sG})$, that is, $G^{\mathfrak {N}}\cap A^{sG}= G^{\mathfrak {N}}\cap A_{sG}$ for every subnormal subgroup A of G.

Keywords

finite groupsoluble groupnilpotent groupnilpotent residual of a groupsubnormal subgroupS-permutable subgroup

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D15: Nilpotent groups, $p$-groups 20D30: Series and lattices of subgroups
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 110 , Issue 3 , December 2024 , pp. 520 - 527
Check for updates
Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

Research of the first and second authors was supported by the National Natural Science Foundation of China (Grant Nos. 12171126, 12101165). Research of the third and fourth authors was supported by the Ministry of Education of the Republic of Belarus (Grant Nos. 20211328, 20211778).

References

Agrawal, R. K., ‘Finite groups whose subnormal subgroups permute with all Sylow subgroups’, Proc. Amer. Math. Soc. 47 (1975), 77–83.CrossRefGoogle Scholar
Ballester-Bolinches, A., Beidleman, J. C. and Heineken, H., ‘Groups in which Sylow subgroups and subnormal subgroups permute’, Illinois J. Math. 47(1–2) (2003), 63–69.CrossRefGoogle Scholar
Ballester-Bolinches, A., Esteban-Romero, R. and Asaad, M., Products of Finite Groups (Walter de Gruyter, Berlin–New York, 2010).CrossRefGoogle Scholar
Ballester-Bolinches, A. and Ezquerro, L. M., Classes of Finite Groups (Springer, Dordrecht, 2006).Google Scholar
Chi, Z. and Skiba, A. N., ‘On a lattice characterisation of finite soluble PST-groups’, Bull. Aust. Math. Soc. 101 (2020), 113–120.CrossRefGoogle Scholar
Deskins, W. E., ‘On quasinormal subgroups of finite groups’, Math. Z. 82 (1963), 125–132.CrossRefGoogle Scholar
Guo, J., Guo, W., Safonova, I. N. and Skiba, A. N., ‘ $G$ -covering subgroup systems for the classes of finite soluble PST-groups’, Comm. Algebra 49(9) (2021), 1–9.CrossRefGoogle Scholar
Guo, W. and Skiba, A. N., ‘Finite groups with given $s$ -embedded and $n$ -embedded subgroups’, J. Algebra 321 (2009), 2843–2860.CrossRefGoogle Scholar
Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin–Heidelberg–New York, 1967).CrossRefGoogle Scholar
Kegel, O. H., ‘Sylow-Gruppen und Subnormalteiler endlicher Gruppen’, Math. Z. 78 (1962), 205–221.CrossRefGoogle Scholar
Miao, L., ‘On weakly $s$ -permutable subgroups of finite groups’, Bol. Soc. Bras. Mat. 41(2) (2010), 223–235.CrossRefGoogle Scholar
Schmid, P., ‘Subgroups permutable with all Sylow subgroups’, J. Algebra 207 (1998), 285–293.CrossRefGoogle Scholar
Skiba, A. N., ‘On weakly $s$ -permutable subgroups of finite groups’, J. Algebra 315(1) (2007), 192–209.CrossRefGoogle Scholar
Wei, H., Lv, Y., Dai, Q., Zhang, H. and Yang, L., ‘Nearly $s$ -embedded subgroups and the $p$ -nilpotency of finite groups’, Comm. Algebra 48(9) (2020), 3874–3880.CrossRefGoogle Scholar