Finite groups whose n-maximal subgroups are σ-subnormal



Let σ = {σ i | iI} 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 σ i -subgroup of G, for some iI, and H contains exactly one Hall σ i -subgroup of G for every σ i 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 HA x = A x H for all AH and xG: σ-subnormal in G if there is a subgroup chain A = A0A1 ≤ · · · ≤ At = G such that either \({A_{i - 1}}\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{ \triangleleft } {A_i}\) or A i =(A i -1)Ai is a finite σ i -group for some σ i σ for all i = 1;:::; t.

If M n < Mn-1 < · · · < M1 < M0 = G, where Mi is a maximal subgroup of Mi-1, i = 1; 2;...; n, then M n 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 (m σq (G) = n, respectively).

In this paper, we show that the parameters m σ (G) and m σq (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 m σ (G) = |П(G)|. Some known results are generalized.


finite group n-maximal subgroup σ-subnormal subgroup σ-quasinormal subgroup σ-soluble group σ-nilpotent group 


20D10 20D20 20D30 20D35 


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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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© Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiChina
  2. 2.Department of Mathematics and Technologies of ProgrammingFrancisk Skorina Gomel State UniversityGomelBelarus

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