Jianhong Huang

Finite groups with given systems of σ-semipermutable subgroups

Let σ = {σi|i ∈ I} be a partition of the set of all primes ℙ and G a finite group. G is said to be σ-soluble if every chief factor H/K of G is a σi-group for some i = i(H/K). 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 σi ∈ σ and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ∩ π(G)≠∅. A subgroup A of G is said to be σ-quasinormal or σ-permutable in G if G has a complete Hall σ-set ℋ such that AHx = HxA for all x ∈ G and all H ∈ℋ. We obtain a new characterization of finite σ-soluble groups G in which σ-permutability is a transitive relation in G.

Finite groups all of whose subgroups are σ-subnormal or σ-abnormal

ABSTRACT Let σ = {σi|i∈I} be a partition of the set of all primes ℙ and G a finite group. 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 exact one Hall σi-subgroup of G for every i such that σi∩π(G)≠∅. A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable or σ-quasinormal in G if G possesses a complete Hall σ-set ℋ such that for all H∈ℋ and all x∈G; σ-subnormal in G if there is a subgroup chain such that either or is σ-primary for all i = 1,…,t; σ-abnormal in G if L∕KL is not σ-primary whenever A≤K<L≤G. In this paper, answering to Question 7.7 in [17], we describe finite groups in which every subgroup is either σ-subnormal or σ-abnormal, and we use this result to classify finite groups G such that every subgroup of G is either σ-quasinormal or σ-abnormal in G.

A covering subgroup system for the class of p-nilpotent groups

Let ℱ be a class of groups and let G be a finite group. We call a set Σ of subgroups of G a covering subgroup system of G for ℱ (or directly an ℱ-covering subgroup system of G) if G ∈ ℱ whenever every subgroup in Σ is in ℱ. We give some covering subgroup systems for the class of all p-nilpotent groups.

On finite groups with generalized σ-subnormal Schmidt subgroups

ABSTRACT Let G be a finite group and σ = {σi|i∈I} some partition of the set of all primes. A subgroup A of G is said to be generalized σ-subnormal in G if A = ⟨L,T⟩, where L is a modular subgroup and T is a σ-subnormal subgroup of G. In this paper, we prove that if every Schmidt subgroup of G is generalized σ-subnormal in G, then the commutator subgroup G′ of G is σ-nilpotent.

On generalized S-quasinormal and generalized subnormal subgroups of finite groups

ABSTRACT Let G be a finite group and H a subgroup of G. We say that H is a generalized subnormal (respectively generalized S-quasinormal) subgroup of G if H = ⟨A,B⟩ for some modular subgroup A and subnormal (respectively S-quasinormal) subgroup B of G. If , where Mi is a maximal subgroup of Mi−1 for all i = 1,…,n, then Mn (n>0) is an n-maximal subgroup of G. In this paper, we study finite groups whose n-maximal subgroups are generalized subnormal or generalized S-quasinormal.

On G-Covering Subgroup Systems for Some Saturated Formations of Finite Groups

Let ℱ be a class of groups and G a finite group. We call a set Σ of subgroups of G a G-covering subgroup system for ℱ if G ∈ ℱ whenever Σ ⊆ ℱ. For a non-identity subgroup H of G, we put Σ H be some set of subgroups of G which contains at least one supplement in G of each maximal subgroup of H. Let p ≠ q be primes dividing |G|, P, and Q be non-identity a p-subgroup and a q-subgroup of G, respectively. We prove that Σ P and Σ P ∪ Σ Q are G-covering subgroup systems for many classes of finite groups.

FINITE GROUPS WITH WEAKLY S-QUASINORMALLY EMBEDDED SUBGROUPS

Let H be a subgroup of a group G. A subgroup H of G is said to be S-quasinormally embedded in G if for each prime p dividing |H|, a Sylow p-subgroup of H is also a Sylow p-subgroup of some S-quasinormal subgroup of G. We say that H is weakly S-quasinormally embedded in G if there exists a normal subgroup T of G such that HT ⊴ G and H ∩ T is S-quasinormally embedded in G. In this paper, we investigate further the influence of weakly S-quasinormally embedded subgroups on the structure of finite groups. A series of known results are generalized.

On generalized m-S-permutable subgroups of a finite group

Abstract In this paper, we give some new conditions under which a normal subgroup E of a finite group G is hypercyclically embedded in G, that is, every chief factor of G below E is cyclic.