Some notes on the rank of a finite soluble group


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Let G be a finite group and let σ = {σi|iI} 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 of subgroups of G is a complete Hall σ-set of G if each member ≠ 1 of is a Hall σi-subgroup of G for some iI and has exactly one Hall σi-subgroup of G for every i such that σiπ(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 such that AHx = HxA for all H and xG. The symbol r(G) (rp(G)) denotes the rank (p-rank) of G.

Assume that is a complete Hall σ-set of G. We prove that (i) if G is soluble, r(H) ≤ r ∈ ℕ for all 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 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 and odd pπ(H).

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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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