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

Abstract

AbstractIf $$\sigma = \{ {\sigma }_{i} : i \in I \}$$ 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.