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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
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
- Type
- Research Article
- Information
- Copyright
- © The Author(s), 2024. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.
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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