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
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.
📖 Papers frequently viewed together
1 Citations
Let σ = {σi|i ∈ I} be some partition of the set ℙ of all primes, that is, ℙ = ∪i∈Iσi and σi ∩ σj = ∅ for all i≠j. We say that a finite group G is σ-soluble if every chief factor H/K of G is a σi-group for some i = i(H/K) ∈ I. We give some characterizations of finite σ-soluble groups.
46 CitationsSource
45 CitationsSource
#1V.A. KovalevaH-Index: 1
#2Xiaolan YiH-Index: 1
5 Citations
#1Viktoria A. Kovaleva (Francisk Skorina Gomel State University)H-Index: 2
A subgroup H of a group G is said to be K-ℙ-subnormal inG [A. F. Vasilyev, T. I. Vasilyeva and V. N. Tyutyanov, On finite groups with almost all K-ℙ-subnormal Sylow subgroups, in Algebra and Combinatorics: Abstracts of Reports of the International Conference on Algebra and Combinatorics on Occasion the 60th Year Anniversary of A. A. Makhnev (Ekaterinburg, 2013), pp. 19–20] if there exists a chain of subgroups H = H0 ≤ H1 ≤ ⋯ ≤ Hn = G such that either Hi-1 is normal in Hi or |Hi : Hi-1| is a prim...
5 CitationsSource
#1Victor S. MonakhovH-Index: 1
#2N. KniahinaH-Index: 1
A subgroup H of a group G is called P-subnormal in G whenever either H = G or there is a chain of subgroups H = H0 ⊂ H1 ⊂ · ·· ⊂ Hn = G such that |Hi : Hi−1| is a prime for all i. In this paper we study groups with P-subnormal 2-maximal subgroups, and groups with P-subnormal primary cyclic subgroups.
14 Citations
#1Li ShirongH-Index: 1
#2Nanning GuangxiH-Index: 1
A subgroup H of a nite group G is called a TI-subgroup if H\H g =1o rH for each g2 G. This paper classies the nite non-nilpotent groups all of whose second maximal subgroups are TI-groups.
20 Citations
Introduction. Dedekind has determined all groups whose subgroups are all normal (see, e.g., [5, Theorem 12.5.4]). Partially generalizing this, Wielandt showed that afinite group is nilpotent, if and only if all its subgroups are subnormal, and also if and only if all maximal subgroups are normal [5, Corollary 10.3.1, 10.3.4]. Huppert [7, Satze 23, 24] has shown that if all 2nd-maximal subgroups of a finite group are normal, the group is supersolvable, while if all 3rd-maximal subgroups are norma...
24 CitationsSource
#1Zvonimir Janko (ANU: Australian National University)H-Index: 7
36 CitationsSource
#1Bertram Huppert (University of Tübingen)H-Index: 13
174 CitationsSource
Cited By5
#1Wenbin Guo (USTC: University of Science and Technology of China)H-Index: 15
#1Guo WenbinH-Index: 1
Last. Alexander N. SkibaH-Index: 20
view all 2 authors...
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...
15 CitationsSource
#1Z. Chi (USTC: University of Science and Technology of China)H-Index: 1
#2Alexander N. Skiba (Francisk Skorina Gomel State University)H-Index: 20
Last. A. N. SkibaH-Index: 1
view all 2 authors...
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 \({\sigma (n)\cap \sigma (m)=\emptyset}\).
3 CitationsSource
#1Ch. Zhang (USTC: University of Science and Technology of China)H-Index: 1
#2A. N. SkibaH-Index: 1
All analyzed groups are finite. Let σ = {σi| i ∈ I} be a partition of the set of all primes ℙ. If n is an integer, then the symbol σ(n) denotes a set {σi| σi ∩ π(n) ≠ ∅}. The integers n and m are called σ -coprime if σ(n) ∩ σ(m) = ∅ . Let t > 1 be a natural number and let 𝔉 be a class of groups. Then we say that 𝔉 is $ {\varSigma}_t^{\sigma } -closed provided that 𝔉 contains each group G with subgroups A1, . . . , At 𝜖 𝔉 whose indices ∣G : A1 ∣ , …, ∣ G : At∣ are pairwise σ -coprime. We...
1 CitationsSource
Let \sigma =\{\sigma_{i} | i\in I\}be some partition of the set of all primes \Bbb{P}and let Gbe a finite group. Then Gis said to be \sigma full if Ghas a Hall \sigma _{i}subgroup for all i A subgroup Aof Gis said to be \sigmapermutable in Gprovided Gis \sigma full and Apermutes with all Hall \sigma _{i}subgroups Hof G(that is, AH=HA for all i We obtain a characterization of finite groups Gin which \sigmapermutability is a transit...
1 Citations
#1Alexander N. SkibaH-Index: 20
This article provides an overview of some recent results and ideas related to the study of finite groups depending on the restrictions on some systems of their sections. In particular, we discuss some properties of the lattice of all subgroups of a finite group related with conditions of permutability and generalized subnormality for subgroups. The paper contains more than 30 open problems which were posed, at different times, by some mathematicians working in the discussed direction.
47 CitationsSource