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On Finite Factorized Groups with \({\mathbb{T}}X\)-Subnormal Subgroups

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Ukrainian Mathematical Journal Aims and scope

Let \({\mathbb{T}}\) be a subset of the set of all natural numbers satisfying the condition

if t ∈ 𝕋, then 𝕋 contains all natural divisors of t. (A)

Recall that a subgroup H is called \({\mathbb{T}}\)-subnormal in G if either H = G or there is a chain of subgroups H = H0 ≤ H1 ≤ . . . ≤ Hn = G such that |Hi : Hi−1| ∈ \({\mathbb{T}}\) for all i. Let X be a normal subgroup of the group G and let \({\mathbb{T}}\) be the set of natural numbers satisfying the condition (A). We introduce the following definition: A subgroup H of the group G is called a \({\mathbb{T}}X\)-subnormal subgroup if H is \({\mathbb{T}}\)-subnormal in HX. Moreover, we study factorizable groups G = AB with \({\mathbb{T}}X\)-subnormal factors A and B. Under additional restrictions imposed on A, B, \({\mathbb{T}}\), and X, we obtain new sufficient conditions for the partial solubility and supersolubility of the analyzed group G.

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Authors and Affiliations

  1. Skorina Gomel University, Gomel, Belarus

    V. S. Monakhov

  2. Pushkin Brest State University, Brest, Belarus

    A. A. Trofimuk

Authors
  1. V. S. Monakhov
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  2. A. A. Trofimuk
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Correspondence to A. A. Trofimuk.

Additional information

Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, No. 10, pp. 1356–1363, October, 2022. Ukrainian DOI: https://doi.org/10.37863/umzh.v74i10.6673.

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Monakhov, V.S., Trofimuk, A.A. On Finite Factorized Groups with \({\mathbb{T}}X\)-Subnormal Subgroups. Ukr Math J 74, 1547–1555 (2023). https://doi.org/10.1007/s11253-023-02154-1

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  • DOI: https://doi.org/10.1007/s11253-023-02154-1

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