Archivo por meses: enero 2022

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.

doi: 10.1016/j.jpaa.2021.106822

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 «Generalised mutually permutable products and saturated formations» published in J. Algebra

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A. Ballester-Bolinches, S. Y. Madanha, M. C. Pedraza-Aguilera.
Generalised mutually permutable products and saturated formations.
J. Algebra, 595:434-443, 2022.

doi: 10.1016/j.jalgebra.2021.12.027

Abstract:

We say that a group G = AB is the weakly mutually permutable product of the subgroups A and B, if A permutes with every subgroup of B containing A ∩ B and B permutes with every subgroup of A containing A ∩ B. We prove that some known results for mutually permutable products remain true for weakly mutually permutable ones. Moreover, if G‘ is nilpotent, A permutes with every Sylow subgroup of B and B permutes with every Sylow subgroup of A, we show that G^F = A^FB^F, where is F a saturated formation containing U, the class of supersoluble groups. This generalises the corresponding result on mutually permutable products.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: weakly mutually permutable products, saturated formations, residuals

Paper «Nilpotent length and system permutability» published in J. Algebra

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Rex Darl, Arnold D. Feldman, M. D. Pérez-Ramos.
Nilpotent length and system permutability.
J. Algebra, 589:287-322, 2022.

Abstract:

If C is a class of groups, a C-injector of a finite group G is a subgroup V of G with the property that V ∩ K is a C-maximal subgroup of K for all subnormal subgroups K of G. The classical result of B. Fischer, W. Gaschütz and B. Hartley states the existence and conjugacy of F-injectors in finite soluble groups for Fitting classes F. We shall show that for groups of nilpotent length at most 4, F-injectors permute with the members of a Sylow basis in the group. We shall exhibit the construction of a Fitting class and a group of nilpotent length 5, which fail to satisfy the result and show that the bound is the best possible.

2020 Mathematics Subject Classification: 20D10, 20D20.

Keywords: Fitting soluble group, Fitting class, injector, system permutability.