Abstract
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 such that either \({A_{i - 1}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } {A_i}\) or Ai=(Ai-1)Ai is a finite σi-group for some σi ∈ σ for all i = 1;:::; t.
If Mn < Mn-1 < · · · < M1 < M0 = G, where Mi is a maximal subgroup of Mi-1, i = 1; 2;...; n, then Mn is said to be an n-maximal subgroup of G. If each n-maximal subgroup of G is σ-subnormal (σ-quasinormal, respectively) in G but, in the case n > 1, some (n−1)-maximal subgroup is not σ-subnormal (not σ-quasinormal, respectively) in G, we write mσ(G) = n (mσq(G) = n, respectively).
In this paper, we show that the parameters mσ(G) and mσq(G) make possible to bound the σ-nilpotent length lσ(G) (see below the definitions of the terms employed), the rank r(G) and the number |П(G)| of all distinct primes dividing the order |G| of a finite soluble group G. We also give the conditions under which a finite group is σ-soluble or σ-nilpotent, and describe the structure of a finite soluble group G in the case when mσ(G) = |П(G)|. Some known results are generalized.
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Acknowledgements
This work was supported by National Nature Science Foundation of China (Grant No. 11771409) and Wu Wen-Tsun Key Laboratory of Mathematics of Chinese Academy of Sciences. The authors thank the referees for their careful reading and helpful comments.
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Guo, W., Skiba, A.N. Finite groups whose n-maximal subgroups are σ-subnormal. Sci. China Math. 62, 1355–1372 (2019). https://doi.org/10.1007/s11425-016-9211-9
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DOI: https://doi.org/10.1007/s11425-016-9211-9
Keywords
- finite group
- n-maximal subgroup
- σ-subnormal subgroup
- σ-quasinormal subgroup
- σ-soluble group
- σ-nilpotent group