# Paper «On σ-subnormality criteria in finite groups» published in J. Pure Appl. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, X. Yi.
On σ-subnormality criteria in finite groups.
J. Pure Appl. Algebra, 226(2):106822, 2022.

Abstract:

Let σ={σ_i: i∈ I} be a partition of the set P of all prime numbers. A subgroup H of a finite group G is called σ-subnormal in G if there is a chain of subgroups H= H_0⊆ H_1⊆⋯⊆ H_n= G where, for every i= 1,…, n, H_{i− 1} normal in H i or H i/Core_{H_i} (H_{i− 1}) is a σ_j-group for some j∈ 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 finite groups are studied.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: finite group, σ-nilpotency, σ-subnormal subgroup.

# Paper «On σ-subnormal subgroups of factorised finite groups» published in J. Algebra

The following paper has been published:
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El següent article ha sigut publicat:

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 : iI} be a partition of the set ℙ of all prime numbers. A subgroup X of a finite group G is called σsubnormal in G if there is chain of subgroups X = X0X1 ⊆⋯⊆ Xn = G with Xj-1 normal in Xj or Xi/CoreXi(Xi-1) is a σ-group for some iI, 1 ≤ jn. 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 gAB , 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