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A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and V. Pérez-Calabuig.
On σ-subnormality criteria in finite σ-soluble groups.
Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114(2):Paper No. 94, 9, 2020.

doi:10.1007/s00009-019-1444-5

Abstract

Let σ = {σ_i : i ∈ I} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σ-subnormal in G if there is a chain of subgroups X = X₀ ⊆ X₁ ⊆⋯⊆ X_n = G where for every j=1,…,n the subgroup X_j-1 is normal in X_j or X_j/Core_Xj(X_j-1) is a σ_i-group for some i ∈ I. In the special case that σ is the partition of ℙ into sets containing exactly one prime each, the σ-subnormality reduces to the familiar case of subnormality. In this paper some σ-subnormality criteria for subgroups of σ-soluble groups, or groups in which every chief factor is a σ_i-group, for some i ∈ I, are showed.

2020 Mathematics Subject Classification: 20D10, 20D20

Keywords: finite group; σ-solubility; σ-nilpotency; σ-subnormal subgroup; factorised group