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A. Ballester-Bolinches, S. F. Kamornikov, M. C. Pedraza-Aguilera, and X. Yi.
On σ-subnormal subgroups of factorised finite groups.
J. Algebra, 559:195–202, 2020.
doi:10.1016/j.jalgebra.2020.05.002
Abstract
Let σ = {σi : 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 chain of subgroups X = X0 ⊆ X1 ⊆⋯⊆ Xn = G with Xj-1 normal in Xj or Xi/CoreXi(Xi-1) is a σ-group for some i ∈ I, 1 ≤ j ≤ n. 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.
If a finite soluble group G = AB is factorised as the product of the subgroups A and B, and X is a subgroup of G such that X is σ-subnormal in 〈X, Xg〉 for all g ∈ A ∪ B , we prove that X is σ-subnormal in G. This is an extension of a subnormality criteria due to Maier and Sidki and Casolo.
2020 Mathematics Subject Classification: 20D10, 20D20
Keywords: Finite group; Soluble group; σ-Subnormal subgroup; σ-Nilpotency; Factorised group