Finite groups whose nmaximal subgroups are \sigmasubnormal

Published on Aug 11, 2016in arXiv: Group Theory
Wenbin Guo15
Estimated H-index: 15
,
Alexander N. Skiba20
Estimated H-index: 20
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Abstract
Let \sigma =\{\sigma_{i} | i\in I\}be some partition of the set of all primes \Bbb{P} A set {\cal H}of subgroups of Gis said to be a \emph{complete Hall \sigma set} of Gif every member \ne 1of {\cal H}is a Hall \sigma_{i}subgroup of G for some i\in I and \cal Hcontains exact one Hall \sigma_{i}subgroup of Gfor every \sigma_{i}\in \sigma (G) A subgroup Hof Gis said to be: \emph{\sigmapermutable} or \emph{\sigmaquasinormal} in Gif Gpossesses a complete Hall \sigmaset set {\cal H}such that HA^{x}=A^{x}Hfor all A\in {\cal H}and x\in G \emph{{\sigma}subnormal} in Gif there is a subgroup chain A=A_{0} \leq A_{1} \leq \cdots \leq A_{t}=Gsuch that either A_{i-1}\trianglelefteq A_{i}or A_{i}/(A_{i-1})_{A_{i}}is a finite \sigma_{i}group for some \sigma_{i}\in \sigmafor all i=1, \ldots t If each nmaximal subgroup of Gis \sigmasubnormal (\sigmaquasinormal, respectively) in Gbut, in the case n > 1 some (n-1)maximal subgroup is not \sigmasubnormal (not \sigmaquasinormal, respectively)) in G we write m_{\sigma}(G)=n(m_{\sigma q}(G)=n respectively). In this paper, we show that the parameters m_{\sigma}(G)and m_{\sigma q}(G)make possible to bound the \sigmanilpotent length \ l_{\sigma}(G)(see below the definitions of the terms employed), the rank r(G)and the number |\pi (G)|of all distinct primes dividing the order |G|of a finite soluble group G We also give conditions under which a finite group is \sigmasoluble or \sigmanilpotent, and describe the structure of a finite soluble group Gin the case when m_{\sigma}(G)=|\pi (G)| Some known results are generalized.
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Let σ = {σi | i ∈ I} be some partition of the set of all primes P. A set H of subgroups of G is said to be a complete Hallσ-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G, for some i ∈ I, and H contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). A subgroup H of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set H such that HAx = AxH for all A ∈ H and x ∈ G: σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ · · · ≤ At = G s...
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Throughout this paper, all groups are finite. \({\sigma =\{\sigma_{i}\mid i\in I \}}\) is some partition of the set of all primes \({\mathbb{P}}\), and \({\sigma (n)= \{\sigma _{i}\mid \sigma _{i}\cap \pi (n)\ne \emptyset \}}\) for any \({n\in \mathbb{N}}\). The natural numbers n and m are called \({\sigma}\)-coprime if \({\sigma (n)\cap \sigma (m)=\emptyset}\).
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