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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]
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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
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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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).

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