On the Normal Subgroup with Exactly Two G-Conjugacy Class Sizes

Let G be a finite group with a non-central Sylow r-subgroup R,Z(G) the center of G,and N a normal subgroup of.G.The purpose of this paper is to determine the structure of N under the hypotheses that N contains R and the G-conjugacy class size of every element of N is either 1 or m.Particularly,it is shown that N is Abelian if N ∩ Z(G)=1 and the Goeonjugacy class size of every element of N is either 1 or m.     

Nilpotent pseudogroups of functions on an interval

A near-identity nilpotent pseudogroup of order m 1 is a family f ₁, . . . , f _n : (-1, 1) of C ² functions for which: for some small positive real number < 1/10 ^m+1 and commutators of the functions f _i of order at least m equal the identity. We present a classification of near-identity nilpotent pseudogroups: our results are similar to those of Plante, Thurston, Farb and Franks. As an application, we classify certain foliations of nilpotent manifolds.     

Bifurcations on a Five-Parameter Family of Planar Vector Field

In this paper we consider a five-parameter family of planar vector fields where μ = (μ ₁, μ ₂, μ ₃, μ ₄, μ ₅), which is a small parameter vector, and c(0) ≠ 0. The family X _μ represents the generic unfolding of a class of nilpotent cusp of codimension five. We discuss the local bifurcations of X _μ, which exhibits numerous kinds of bifurcation phenomena including Bogdanov-Takens bifurcations of codimension four in Li and Rousseau (J. Differ. Eq. 79, 132–167, 1989) and Dumortier and Fiddelaers (In: Global analysis of dynamical systems, 2001), and Bogdanov-Takens bifurcations of codimension three in Dumortier et al. (Ergodic Theory Dynam. Syst. 7, 375–413, 1987) and Dumortier et al. (Bifurcations of planar vector fields. Nilpotent singularities and Abelian integrals, 1991). After making some rescalings, we obtain the truncated systems of X _μ . For a truncated system, all possible bifurcation sets and related phase portraits are obtained. When the truncated system is a Hamiltonian system, the bifurcation diagram and the related phase portraits are given too. Hopf bifurcations are studied for another truncated system. And it shows that the system has the Hopf bifurcations of codimension at most three, and at most three limit cycles occur in the small neighborhood of the Hopf singularity. Dedicated to Professor Zhifen Zhang in the occasion of her 80th birthday     

极小子群对有限群结构的影响

设Ｇ是一个有限群．Ｇ的极小子群如何影响群的结构是一个人们感兴趣的问题．在本文中,我们用极小子群的ｃ－正规的条件刻划群Ｇ的结构．我们推广了一些已知的结果．     

李三系的Frattini子系

李三系是从黎曼对称空间产生的三元运算的代数系统,近年来备受数学家们的重视.针对李三系的Frattini子系和基本李三系的问题进行了研究,给出了Frattini子系和基本李三系的一些性质,并证明了李三系的非嵌入定理,同时得到了幂零李三系是基本李三系的一个充要条件.     

带有限性条件Abel群的自同态环和自同构群

给出了带极大或极小条件的Abel群A的自同构群以及自同态环的相伴Lie环是可解或幂零的充要条件.同时也给出了群A=Q_(π1)⊕Q_(π2)⊕…⊕Q_(πr)的自同构群是可解或幂零的充要条件,以及群A的自同态环的相伴Lie环是可解或幂零的充要条件.     

Finitely Generated Nilpotent Groups of Infinite Cyclic Commutator Subgroups

The aim of this paper is to determine the structure and to establish the isomorphic invariant of the finitely generated nilpotent group G of infinite cyclic commutator subgroup. Using the structure and invariant of the group which is the central extension of a cyclic group by a free abelian group of finite rank of infinite cyclic center, we provide a decomposition of G as the product of a generalized extraspecial Z-group and its center. By using techniques of lifting isomorphisms of abelian groups and equivalent normal form of the generalized extraspecial Z-groups, we finally obtain the structure and invariants of the group G.     

幂零Fuzzy矩阵的一些结论及幂零度的求法

本文给出计算幂零Fuzzy矩阵的幂零度的两个办法,同时给出一个有有向Hamilton路的有向图的一个充分条件.