On σ-semipermutable Subgroups of Finite Groups

Let a = {σi｜ i ∈ I} be some partition of the set of all primes P, G a finite group and σ（G） = {σi｜σi ∩ π （G） ≠ Ф}. A set H of subgroups of G is said to be a complete Hall or-set of G if every member ≠ 1 of H is a Hall σi-subgroup of G for some σi ∈ σ 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-semipermutablc in G with respect to H if HHi x = Hi x H for all x ∈ G and all x ∈ G and all Hi ∈H such that （｜H｜, ｜Hi｜） = 1; σ-semipermutable in G if H is σ-semipermutable in G with respect to some complete Hall σ-set of G. We study the structure of G being based on the assumption that some subgroups of G are σ-semipermutable in G.

A Generalization of σ-Permutability

Throughout this paper,all groups are finite and G always denotes a finite group;σis some partition of the set of all primes P.A group G is said to beσ-primary if G is aπ-group for someπ∈σ.Aπ-semiprojector of G[29]is a subgroup H of G such that HN/N is a maximalπ-subgroup of G/N for all normal subgroups N of G.LetП⊆σ.Then we say thatχ={X_(1),...,X_(t)}is aП-covering subgroup system for a subgroup H in G if all members of the setχareσ-primary subgroups of G and for eachπ∈Пwithπ∩π(H)≠φthere are an index i and aπ-semiprojector U of H such that U≤X_(i).We study the embedding properties of subgroups H of G under the hypothesis that G has aП-covering subgroup systemχsuch that H permutes with X^(x)for all X∈χand x∈G.Some well-known results are generalized.

Finite groups whose n-maximal subgroups are σ-subnormal

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 Hall σ-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 Ai-1■Ai 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 |π(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)=|π(G)|. Some known results are generalized.