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A CHARACTERISATION OF SOLUBLE
${PST}$-GROUPS
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 110 / Issue 3 / December 2024
- Published online by Cambridge University Press:
- 15 March 2024, pp. 520-527
- Print publication:
- December 2024
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- Article
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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.
