Finite groups with σ-subnormal Schmidt subgroups

If $$\sigma = \{ {\sigma }_{i} : i \in I \}$$ σ = { σ i : i ∈ I } is a partition of the set $$\mathbb {P}$$ P of all prime numbers, a subgroup H of a finite group G is said to be $$\sigma $$ σ -subnormal in G if H can be joined to G by means of a chain of subgroups $$H=H_{0} \subseteq H_{1} \subseteq \cdots \subseteq H_{n}=G$$ H = H 0 ⊆ H 1 ⊆ ⋯ ⊆ H n = G such that either $$H_{i-1}$$ H i - 1 normal in $$H_{i}$$ H i or $$H_{i}/{{\,\mathrm{Core}\,}}_{H_{i}}(H_{i-1})$$ H i / Core H i ( H i - 1 ) is a $${\sigma }_{j}$$ σ j -group for some $$j \in I$$ j ∈ I , for every $$i=1, \ldots , n$$ i = 1 , … , n . If $$\sigma = \{\{2\}, \{3\}, \{5\}, ... \}$$ σ = { { 2 } , { 3 } , { 5 } , . . . } is the minimal partition, then the $$\sigma $$ σ -subnormality reduces to the classical subgroup embedding property of subnormality. A finite group X is said to be a Schmidt group if X is not nilpotent and every proper subgroup of X is nilpotent. Every non-nilpotent finite group G has Schmidt subgroups and a detailed knowledge of their embedding in G can provide a deep insight into its structure. In this paper, a complete description of a finite group with $$\sigma $$ σ -subnormal Schmidt subgroups is given. It answers a question posed by Guo, Safonova and Skiba.

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“…The role of the σ -Fitting subgroup in the proof of Theorem 2 is determined by the main result of [11].…”

Section: (G)) In Particular S Is a Normal Subgroup Of S O π (G)mentioning

confidence: 99%

Finite Groups with $$\sigma $$-Subnormal Schmidt Subgroups

Ballester‐Bolinches

Kamornikov

2022

Bull. Malays. Math. Sci. Soc.

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“…The structure of Schmidt groups is well known (see [10, III, Satz 5.2] and [2]) and such groups have deep applications in the theory of the classes of groups [3, 8]. Groups in which the condition of subnormality or generalised subnormality is satisfied for all or selected Schmidt subgroups are studied in [12, 17] and the recent papers [1, 9, 11, 13, 15, 19]. In this article, we consider, in a certain sense, the opposite situation.…”

Section: Introductionmentioning

confidence: 99%

“…A subgroup H of G is said to be abnormal in G if for all . From the results in [1, 9, 11, 13, 15, 19], it is natural to ask: What is the structure of a group in which all Schmidt subgroups are abnormal? We provide an answer to this question.…”

Section: Introductionmentioning

confidence: 99%