FINITE p-GROUPS WHICH CONTAIN A SELF-CENTRALIZING CYCLIC NORMAL SUBGROUP

For any prime p, all finite noncyclic p-groups which contain a self-centralizing cyclic normal subgroup are determined by using cohomological techniques. Some applications are given, including a character theoretic description for such groups.

On Finite Groups that are the Product of Two Subnormal Supersoluble Subgroups

Denote the class of finite groups that are the product of two normal supersoluble subgroups and the class of groups that are the product of two subnormal supersoluble subgroups by B_(1)and B_(2),respectively.In this paper,a characterisation of groups in B_(1)or in B_(2)is given.By applying this new characterisation,some new properties of B_(1)(B_(2))and new proofs of many known results about B_(1)or B_(2)are obtained.Further,closure properties of B_(1)and B_(2)are discussed.

关于一道抽象代数题的推广

用初等的方法证明了:设G是有限群,p是G的最小素因子,若对任意n∣|G|都有|{x∈G∣x^(n)=1}|≤(p-1)n,则G是循环群.推广了教材[1]第1.4节的习题3.

p-超可解群的新判定

设p是有限群G的一个素因子,P是G的p-子群且t是正整数满足|P|>p^(t).通过P的任意pt阶子群和4阶循环子群(P为非交换2-群且t=1)在G中c-正规或者Φ-可补,得到p-超可解群的一些新判定方法.

A Characterization of L_n(q) by the Normalizers' Orders of Their Sylow Subgroups

Let G be a finite group. If |NG（R1）|=|NLn（q）（R2）| for every prime r, where R1∈Sylr G and R2∈Sylr（Ln（q））, then G≌Ln（q）.

A characterization of the finite simple groups with set of indices of their maximal subgroups

We study the problem concerning the influence of indices of maximal subgroups of a simple group on the structure of a group. We obtain a characterization property of all finite simple groups.

Finite p-Groups with a Class of Complemented Normal Subgroups

Assume G is a finite group and H a subgroup of G. If there exists a subgroup K of G such that G = HK and H ∩ K = 1, then K is said to be a complement to H in G. A finite p-group G is called an NC-group if all its proper normal subgroups not contained in de（G） have complements. In this paper, some properties of NC-groups are investigated and some classes of NC-groups are classified. Keywords Finite p-groups, normal subgroups, subgroup complement

On the Index of Finite Determinacy of Vector Fields with Resonances

In this paper we study normal forms for a class of germs of 1-resonant vector fields on R^n with mutually different eigenvalues which may admit extraneous resonance relations. We give an estimation on the index of finite determinacy from above as well as the essentially simplified polynomial normal forms for such vector fields. In the case that a vector field has a zero eigenvalue, the result leads to an interesting corollary, a linear dependence of the derivatives of the hyperbolic variables on the central variable.

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.

关于有限群的非循环真子群的自正规性和非正规性

为了进一步讨论非循环真子群的自正规性和非正规性之间的联系,通过分析相关的群结构,给出了一个直接的证明得到了有限群G的每个非循环真子群皆自正规当且仅当G为每个非循环真子群皆非正规的可解群。

Local symmetries of finite type hypersurfaces in C^2 Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

The first part of this paper gives a complete description of local automorphism groups for Levi degenerate hypersurfaces of finite type in C2. It is also proved that, with the exception of hypersurfaces of the form v = |z|k, local automorphisms are always determined by their 1-jets. Using this result, the second part describes special normal forms which by an additional normalization eliminate the nonlinear symmetries of the model and allows to decide effectively about local equivalence of two hypersurfaces given in this normal form.

A Note on TI-Subgroups of a Finite Group

Let G be a finite group and H a subgroup of G. Recall that H is said to be aTI-subgroup ofG ifHg∩H = 1 or H for each b∈ G. In this note, we prove that if all non-nilpotent subgroups of a finite non-nilpotent group G are TI-subgroups, then G is soluble, and all non-nilpotent subgroups of G are normal.

A quantitative characterization of finite simple groups

In this note, all groups are finite and the notations and termentary are standard, inaddition, given a finite G group. let π(G) and π(|G|) denote the set of primes dividing |G|.

Control of Wave Propagation and Effect of Kerr Nonlinearity on Group Index

We use four-level atomic system and control the wave propagation via forbidden decay rate. The Raman gain process becomes dominant on electromagnetically induced transparency (EIT) medium by increasing the forbidden decay rate via increasing the number of atoms [G.S. Agarwal and T.N. Dey, Phys. Rev. A 74 (2005) 043805 and K. Harada, T. Kanbashi, and M. Mitsunaga, Phys. Rev. A 73 (2006) 013803]. The behavior of wave propagation is dramatically changed from normal (subluminal) to anomalous (superluminal) dispersion by increasing the forbidden decay rate. The system can also give a control over the group velocity of the light propagating through the medium via Kerr field.

非正规子群共轭类数为p+2的有限p群的分类

利用非正规子群的共轭类研究有限p群的结构是有限p群领域一个重要的研究内容,其中p表示一个素数.对于非正规子群共轭类数为p+2的有限p群,按照导群的阶分p和p^(2)两种情况进行讨论,利用中心积和中心扩张的方法,给出了非正规子群共轭类数为p+2的有限p群的具体分类.

Finite p-groups whose non-normal subgroups have few orders

Suppose that G is a finite p-group. If G is not a Dedekind group, then G has a non-normal subgroup. We use p^M（G） and p^m（G） to denote the maximum and minimum of the orders of the non-normal subgroups of G, respectively. In this paper, we classify groups G such that M（G） 〈 2m（G） ^- 1. As a by-product, we also classify p-groups whose orders of non-normal subgroups are p^k and p^k＋1.

On c#-normal subgroups in finite groups

A subgroup H of a finite group G is called a c#-normal subgroup of G if there exists a normal subgroup K of G such that G = HK and H n K is a CAP-subgroup of G. In this paper, we investigate the influence of fewer c#- normal subgroups of Sylow p-subgroups on the p-supersolvability, p-nilpotency, and supersolvability of finite groups. We obtain some new sufficient and necessary conditions for a group to be p-supersolvable, p-nilpotent, and supersolvable. Our results improve and extend many known results.

On Splitting of the Normalizer of a Maximal Torus in E6(q)

Let G be a finite group of Lie type E6 over Fq (adjoint or simply connected) and W be the Weyl group of G. We describe maximal tori T such that T has a complement in its algebraic normalizer N(G, T). It is well known that for each maximal torus T of G there exists an element w ∈ W such that N(G, T )/T ■ CW(w). When T does not have a complement isomorphic to CW(w), we show that w has a lift in N(G, T) of the same order.

Finite Groups Whose Nontrivial Normal Subgroups Have Order Two

In this paper, we investigate the structure of the groups whose nontrivial normal subgroups have order two. Some properties of this kind of groups are obtained.

3p^2阶连通4度Cayley图的正规性

对于一类3p2(p是素数)阶群G=〈a,b,c ap=bp=c3=[a,b]=1,-c 1ac=a-rbr+1,-c 1bc=a,r>1,r3≡1(m od p)〉,研究了其连通4度C ay ley图的正规性,并通过其点稳定子的结构证明G的连通4度C ay ley图均正规。鉴于王艳丽等人的相关工作,这等于圆满解决了3p2阶群的连通4度C ay ley图的正规性问题。