Some notes on the rank of a finite soluble group
- 作者: Zhang L.1, Guo W.1, Skiba A.N.1
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隶属关系:
- Department of Mathematics
- 期: 卷 58, 编号 5 (2017)
- 页面: 915-922
- 栏目: Article
- URL: https://journals.rcsi.science/0037-4466/article/view/171519
- DOI: https://doi.org/10.1134/S0037446617050196
- ID: 171519
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Let G be a finite group and let σ = {σi|i ∈ I} be some partition of the set ℙ of all primes. Then G is called σ-nilpotent if G = A1 × ⋯ × Ar, where Ai is a \({\sigma _{{i_j}}}\)-group for some ij = ij(Ai). A collection ℋ of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of ℋ is a Hall σi-subgroup of G for some i ∈ I and ℋ has exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G) ≠ ø. A subgroup A of G is called σ-quasinormal or σ-permutable [1] in G if G possesses a complete Hall σ-set ℋ such that AHx = HxA for all H ∈ ℋ and x ∈ G. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G.
Assume that ℋ is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all H ∈ ℋ, and every n-maximal subgroup of G (n > 1) is σ-quasinormal in G, then r(G) ≤ n+r − 2; (ii) if every member in ℋ is soluble and every n-minimal subgroup of G is σ-quasinormal, then G is soluble and rp(G) ≤ n + rp(H) − 1 for all H ∈ ℋ and odd p ∈ π(H).
作者简介
L. Zhang
Department of Mathematics
编辑信件的主要联系方式.
Email: zhang12@mail.ustc.edu.cn
中国, Hefei
W. Guo
Department of Mathematics
Email: zhang12@mail.ustc.edu.cn
中国, Hefei
A. Skiba
Department of Mathematics
Email: zhang12@mail.ustc.edu.cn
白俄罗斯, Minsk
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