On σ -Residuals of Subgroups of Finite Soluble Groups

A. A. Heliel, A. Ballester-Bolinches (), Mohammed Al-Shomrani and R. A. Al-Obidy
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A. A. Heliel: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
A. Ballester-Bolinches: Departament de Matemàtiques, Universitat de València, Dr. Moliner, 50, Burjassot, 46100 Valencia, Spain
Mohammed Al-Shomrani: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia
R. A. Al-Obidy: Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia

Mathematics, 2023, vol. 11, issue 10, 1-7

Abstract: Let σ = { σ i : i ∈ I } be a partition of the set of all prime numbers. A subgroup H of a finite group G is said to be σ - subnormal in G if H can be joined to G by a chain of subgroups H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G where, for every j = 1 , ⋯ , n , H j − 1 is normal in H j or H j / C o r e H j ( H j − 1 ) is a σ i -group for some i ∈ I . Let B be a subgroup of a soluble group G normalising the N σ -residual of every non- σ -subnormal subgroup of G , where N σ is the saturated formation of all σ -nilpotent groups. We show that B normalises the N σ -residual of every subgroup of G if G does not have a section that is σ -residually critical.

Keywords: finite group; formation; residual; ?-subnormality (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2023
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