# Finite groups whose n-maximal subgroups are σ-subnormal

Articles

## Abstract

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.

## Keywords

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

## MSC(2010)

20D10 20D20 20D30 20D35

## Notes

### 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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