Finite groups whose n-maximal subgroups are σ-subnormal

Let sigma = {sigma(i) | 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 Hall sigma-set of G if every member 1 of H is a Hall sigma(i)-subgroup of G, for some i I, and H contains exactly one Hall sigma(i)-subgroup of G for every sigma(i) sigma(G). A subgroup H of G is said to be: sigma-permutable or sigma-quasinormal in G if G possesses a complete Hall sigma-set H such that HA(x) = A(x)H for all A H and x G: sigma-subnormal in G if there is a subgroup chain A = A(0) A(1) A(t) = G such that either Ai-1Ai or A(i)=(A(i)-1)Ai is a finite sigma(i)-group for some sigma(i) sigma for all i = 1;:::; t.If M-n < Mn-1 < < M-1 < M-0 = 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 sigma-subnormal (sigma-quasinormal, respectively) in G but, in the case n > 1, some (n-1)-maximal subgroup is not sigma-subnormal (not sigma-quasinormal, respectively) in G, we write m sigma(G) = n (m(sigma q)(G) = n, respectively).In this paper, we show that the parameters m(sigma)(G) and m(sigma q)(G) make possible to bound the sigma-nilpotent length l sigma(G) (see below the definitions of the terms employed), the rank r(G) and the number |<SIC>(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 sigma-soluble or sigma-nilpotent, and describe the structure of a finite soluble group G in the case when m(sigma)(G) = |<SIC>(G)|. Some known results are generalized.