S-半正规性与P-幂零性

In this paper, we deal mainly with the following problem: if every 2-maximal subgroup of a Sylow p-subgroup of a finite group G is S-seminormal in G, what conditions force G to be p-nilpotent? As an application of main results, some sufficient conditions for finite nilpotent groups and finite supersolvable groups are obtained.     

关于有限群两个超可解子群之积的问题

In this paper, we give some sufficient conditions for products of two supersolvable sub-groups to be supersolvable groups. Our results generalize some known results.Theorem 1 Let G = HK,(|H|,|K|) = 1, Where H and K are two supersolvable sub-groups. If H is commutative with every maximal subgroup of K, and K is commutative with every maximal subgroup of H, then G is supersolvable.Theorem 2 Let G = HK, H ∩ K = 1, H G, and K be quasinormal in H. If H, K are supersolvable, the G is supersolvable.Theorem 3 Let G= HK,(|H|,|K|) = 1,H,K be two supersolvable subgroups. If H is commutative with any Sylow subgroup of K and any maximal subgroup of every sylow subgroup of K, and K is commutative with any sylow subgroup of H and any maximal subgroup of every sylow subgroup of H, then G is supersolvable. Theorem 4 If H,K are two supersolvable subgroups of G, G= HK, G′is nilpotent, H is quasi normal K, and K is quasi normal in H,then G is supersolvable. Theorem 5 If H,K are two supersolvable subgroups of G, G= HK, H′? G,[H,K]? G,[H,K] is nilpotent, H is quasi normal in K, and K is quasi normal in H,then G is supersolvable.     

A generalized CAP-subgroup of a finite group

Let A be a subgroup of a finite group G. We say that A is a generalized CAP-subgroup of G if for each chief factor H/K of G either A avoids H/K or the following holds:(1) If H/K is non-abelian, then|H :(A ∩H)K | is a p′-number for every p ∈π((A ∩H)K/K);(2) If H/K is a p-group, then |G : NG(K(A ∩H))| is a p-number. In this paper, we use the generalized CAP-subgroup to characterize the structure of finite groups.Some new characterizations of the hypercyclically embedded subgroups of a finite group are obtained and a series of known results are generalized.     

关于F-反正规子群的θ*-完备（英文）

Let F be a saturated formation of finite groups. Given a proper subgroup H of a group G, a subgroup H of G is called F-normal in G if G/HGbelongs to F; otherwise H is said to be F-abnormal in G. In this paper, we investigate the structure of a finite group by θ*-completions of F-abnormal subgroups.     

On θ*-completions of F-abnormal Subgroups

Let F be a saturated formation of finite groups. Given a proper subgroup H of a group G, a subgroup H of G is called F-normal in G if G/HG belongs to F;otherwise H is said to be F-abnormal in G. In this paper, we investigate the structure of a finite group byθ?-completions of F-abnormal subgroups.     

On s-quasinormal and c-normal subgroups of a finite group

Let F be a saturated formation containing the class of supersolvable groups and let G be a finite group. The following theorems are shown： （1） G ∈ F if and only if there is a normal subgroup H such that G/H ∈ F and every maximal subgroup of all Sylow subgroups of H is either c-normal or s-quasinormally embedded in G; （2） G ∈F if and only if there is a soluble normal subgroup H such that G/H∈F and every maximal subgroup of all Sylow subgroups of F（H）, the Fitting subgroup of H, is either e-normally or s-quasinormally embedded in G.     

On Non-commuting Sets in a Finite p-group with Derived Subgroup of Prime Order

Let G be a finite group. A nonempty subset X of G is said to be noncommuting if xy≠yx for any x, y ∈ X with x≠y. If |X| ≥ |Y| for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite p-group with derived subgroup of prime order.     

On s-semipermutable subgroups of finite groups

Suppose that G is a finite group and H is a subgroup of G. We say that H is ssemipermutable in G if HGv = GpH for any Sylow p-subgroup Gp of G with （p, ｜H｜） = 1. We investigate the influence of s-semipermutable subgroups on the structure of finite groups. Some recent results are generalized and unified.     

On σ-semipermutable Subgroups of Finite Groups

Let σ = {σ_i|i ∈ I } be some partition of the set of all primes P, G a finite group andσ(G) = {σ_i |σ_i ∩π(G) = ?}. 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 σ_i-subgroup of G for some σ_i ∈σ and H contains exactly one Hallσ_i-subgroup of G for every σ_i ∈σ(G). A subgroup H of G is said to be: σ-semipermutable in G with respect to H if H H_i~x= H_i~xH for all x ∈ G and all H_i ∈ H such that(|H|, |H_i|) = 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.     

On Minimal Degrees of Faithful Characters of Finite Groups With a T.I.Sylow p-subgroup

The finite group G is said to have T.I. Sylow p-subgroup P, if two differentconjugats of P have only the identity element in common. Using the classificationof the finite simple groups, T.R. Berger, P. Landrock and G.O. Michler provedthe following theorem in 1985, which was conjectured to hold by H.S.Leonard in1968. Theorem 1. Let G be a finite group with a T.I. Sylow p-subgroup P. If Ghas a faithful complex character X such that X(1)≤|P|~(1/2)-1,then P is normalin G.     

有限群的最大子群的性质对群结构的影响

有限群G的一个子群称为在G中是π-拟正规的若它与G的每一个Sylow-子群是交换的.G的一个子群H称为在G中是c-可补的若存在G的子群N使得G=HN且H∩N≤HG=CoreG(H).本文证明了:设F是一个包含超可解群系u的饱和群系,G有一个正规子群H使得G/H∈F.则G∈F若下列之一成立:(1)H的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的;(2)F*(H)的每个Sylow子群的所有极大子群在G中或者是π-拟正规的或者是c-可补的,其中F*(H)是H的广义Fitting子群.此结论统一了一些最近的结果.     

p-超循环嵌入子群的一个判别准则

令E是有限群G的一个正规子群,且U是所有有限超可解群的集合.E称为在G中是p-超循环嵌入的,如果E的每个pd-阶的G-主因子是循环的.G的子群H称为在G中是U-Φ-可补充的,如果存在G的一个次正规子群T,使得G=HT,且(H∩T)H_G/H_G≤Φ/(H/H_G)Z_U(G/H_G),其中Z_U(G/H_G)是商群G/H_G的U-超中心.作者证明,如果E的一些p-子群在G中是U-Φ-可补充的,那么E在G中是p-超循环嵌入的.作为应用,得到了有限群是p-超可解的若干判断准则,并且推广了一些已知的结果.     

有限群的弱正规子群与p-幂零性

研究了p2阶子群以及一般的pk阶子群为弱正规子群时有限群G的结构.给出了有限群为p-幂零群以及超可解群的一些条件.     

关于有限群的Pronormal极小子群

利用有限群G的pronormal极小子群和Sylow子群正规化子中的素数阶弱左Engel元素得到了G成为p-幂零群、幂零群和超可解群的一些充分条件,这些结果推广了已知结论。     

关于一类极大子群的θ-偶

对于有限群G的极大子群M，令β（G：M）表示整除│G：M│的素因子个数，β（G）表示所有β（G；M）中的最大数．令μ（G）为使得β（G：M）=β（G）的极大子群的集合．通过对这一类极大子群的θ-偶赋予一定条件，得到了判断群G可解、超可解的新结果．     

On F-Abnormal Maximal Subgroups of Finite Groups

For any saturated formation F of finite groups containing all supersolvable groups, the groups in F are characterized by F-abnormal maximal subgroups.     

On $\mathfrak{F}_\tau$-$s$-Supplemented Subgroups of Finite Groups

Let $\mathfrak{F}$ be a non-empty formation of groups, $\tau$ a subgroup functor and $H$ a $p$-subgroup of a finite group $G.$ Let $\overline{G}=G/H_G$ and $\overline{H} =H/H_G.$ We say that $H$ is $\mathfrak{F}_\tau$-$s$-supplemented in $G$ if for some subgroup $\overline{T}$ and some $\tau$-subgroup $\overline{S}$ of $\overline{G}$ contained in $\overline{H},$ $\overline{H}\overline{T}$ is subnormal in $\overline{G}$ and $\overline{H} ∩ \overline{T} ≤ \overline{S}Z_{\mathfrak{F}}(\overline{G}).$ In this paper, we investigate the influence of $\mathfrak{F}_\tau$-$s$-supplemented subgroups on the structure of finite groups. Some new characterizations about solubility of finite groups are obtained.     

有限群的超可解性

本文利用子群格的模的性质对有限群进行研究 ,得到了有限超可解群的若干充分条件 ,推广了许多已有结果 .     

弱SS-半置换性极小子群对有限群超可解性的影响

引入弱SS-半置换子群的概念,介绍了弱SS-半置换子群的性质,结合有限群G的极小于群的弱SS-半置换性,并结合C-正规性来讨论有限群的超可解性及幂零性,得到了有限群超可解及幂零的若干充分或充要条件,同时推广了某些著名结果.