Paper «On the Kegel–Wielandt σ‐problem for binary partitions» published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. F. Kamornikov, V. N. Tyutyanov
On the Kegel–Wielandt σ‐problem for binary partitions.
Ann. Mat. Pura Appl., 201:443-451, 2022.

doi: 10.1007/s10231-021-01123-4

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= X_0⊆ X_1⊆⋯⊆ X_n= G where, for every i= 1,…, n, the subgroup X_{i− 1} normal in X_ i or X_ i/Core_{X_i} (X_{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. A finite group G is σ-complete if G possesses at least one Hall σ i -subgroup for every i ∈ I , and a subgroup H of G is said to be σ_i-subnormal in G if H ∩ S is a Hall σ_i-subgroup of H for any Hall σ_i-subgroup S of G. Skiba proposes in the Kourovka Notebook the following problem (Question 19.86), that is called the Kegel–Wielandt σ-problem: Is it true that a subgroup H of a σ-complete group G is σ-subnormal in G if H is σ_i-subnormal in G for all i ∈ I? The main goal of this paper is to solve the Kegel–Wielandt σ-problem for binary partitions.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: Finite group; Hall subgroup; σ-subnormal subgroup; factorised group

Paper «On a problem of L.A. Shemetkov on superradical formations of finite groups» published in J. Algebra

The following paper has been published.

El siguiente artículo ha sido publicado.

El següent article ha sigut publicat.

A. Ballester-Bolinches, S. F. Kamornikov, V. N. Tyutyanov

On a problem of L.A. Shemetkov on superradical formations of finite groups

J. Algebra, 403, 69-76 (2014)

http://dx.doi.org/10.1016/j.jalgebra.2014.01.004

Abstract: Subgroup-closed saturated formations F which are closed under taking products of F-subnormal F-subgroups are studied in the paper. Our results can be regarded as further developments in the hunt for a solution of a problem proposed by L.A. Shemetkov in 1999 in the Kourovka Notebook.

MSC: 20D10, 20D20
Keywords: Finite group, saturated formation, critical group