有限群的s-条件置换子群

如果对群G的任意Sylow子群T,存在一个元素x∈G,使得HT~x=T~xH,那么称群G的子群H在G中s-条件置换．利用s-条件置换子群给出了一些群的性质和结构．     

Globally Irreducible Representations of Finite Groups and Integral Lattices

The notion of globally irreducible representations of finite groups has been introduced by B. H. Gross, in order to explain new series of Euclidean lattices discovered recently by N. Elkies and T. Shioda using Mordell--Weil lattices of elliptic curves. In this paper we first give a necessary condition for global irreducibility. Then we classify all globally irreducible representations of L ₂(q) and ²B₂(q), and of the majority of the 26 sporadic finite simple groups. We also exhibit one more globally irreducible representation, which is related to the Weil representation of degree (pⁿ-1)/2 of the symplectic group Sp_2n(p) (p 1 (mod 4) is a prime). As a consequence, we get a new series of even unimodular lattices of rank 2(pⁿ–1). A summary of currently known globally irreducible representations is given.     

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.     

Maximal Subgroups of Finite Orthogonal Groups Stabilizing Spreads of Lines

We prove that, for q odd and n ≥ 3, the group G = O _n (q ²) · 2 is maximal in either the orthogonal group O _2n (q) or the special orthogonal group SO _2n (q). The group G corresponds to the stabilizer of a spread of lines of PG(2n ? 1, q) in which some lines lie on a quadric, some are secant to the quadric, and others are external to the quadric.     

Quasi-Invariant Measures and Irreducible Representations of the Inductive Limit of Special Linear Groups

Unitary representations of the group are constructed. The construction uses G-quasi-invariant measures on some G-spaces that are subspaces of the space of two-way infinite real matrices. We give a criterion for the irreducibility of these representations.     

Random Walks on Wreath Products of Groups

We bound the rate of convergence to uniformity for a certain random walk on the complete monomial groups GS _n for any group G. Specifically, we determine that n log n+ n log (|G|–1|) steps are both necessary and sufficient for ² distance to become small. We also determine that n log n steps are both necessary and sufficient for total variation distance to become small. These results provide rates of convergence for random walks on a number of groups of interest: the hyperoctahedral group ₂S _n , the generalized symmetric group _m S _n , and S _m S _n . In the special case of the hyperoctahedral group, our random walk exhibits the cutoff phenomenon.     

子群次正规性对有限群可解性的影响

考查了次正规子群对有限群结构的影响,得到有限群可解的若干充分条件和超可解的一个充分条件.     

On the Solvability of Finite Irreducible Linear Groups with Hall TI-Subgroups

Conditions for a -solvable complex linear group of a relatively small degree to be solvable are found.     

The Influence of Primitive Subgroups on the Structure of Finite Groups

A subgroup $H$ of a group $G$ is said to be primitive if it is a proper subgroup of the intersection of all subgroups of $G$ containing $H$ as its proper subgroup. The purpose of this note is to go further into the influence of primitive subgroups on the structure of finite groups. Some new results are obtained.     

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.     

The Influence of Some Quantitative Properties of Abelian Subgroups on Solvability of Finite Groups

As an important application of Thompson's theorem [9 Robinson , D. J. S. ( 1996 ). A Course in the Theory of Groups. , 2nd ed. New York : Springer-Verlag .[Crossref] , [Google Scholar], Theorem 10.4.2], a finite group is solvable if it has an abelian maximal subgroup. In this article, we mainly investigate the influence of some quantitative properties of abelian subgroups on solvability of finite groups. Some new results are obtained.     

Weak Mixing of Random Walks on Groups

Let G be a locally compact -compact group with right Haar measure m and a regular probability measure on G. We say that is weakly mixing if for all gL (G) and all fL ¹(G) with fdm=0 we have n ^{–1 n} _k=1| ^k *f,g|0. We show that is weakly mixing if and only if is ergodic and strictly aperiodic. To prove this we use and prove some results about unimodular eigenvalues for general Markov operators.     

On the Average Number of Unitary Factors of Finite Abelian Groups (II)

Let t(G) be the number of unitary factors of finite abelian group G. In this paper we prove T(x)=∑ _|G|≤x t(G) = main terms for any exponent pair (κ1/2 + 2κ), which improves on the exponent 9/25 obtained by Xiaodong Cao and the author. Received December 8, 1998, Revised April 27, 1998, Accepted June 12, 1998     

Estimate of the L^p-Fourier Transform on Strong ＊-Regular Exponential Solvable Norm Lie Groups

We study the L^p-Fourier transform for a special class of exponential Lie groups, the strong ＊-regular exponential Lie groups. Moreover, we provide an estimate of its norm using the orbit method.     

Introduction to Actions of Discrete Groups on Pseudo-Riemannian Homogeneous Manifolds

Isometric actions of discrete groups are not always properly discontinuous for pseudo-Riemannian manifolds. This short exposition gives an up-to-date survey of some of the basic questions about discontinuous groups for pseudo-Riemannian homogeneous spaces, on which a rapid development has been made since late 1980s.The first half includes an elementary geometric motivation, the Calabi–Markus phenomenon, the discontinuous dual, and deformation. These topics are rebuilt on a criterion of properly discontinuous actions on homogeneous spaces of reductive groups, obtained by Kobayashi [Math. Ann. 1989] and generalized independently by Benoist [Ann. Math. 1996] and Kobayashi [J. Lie Theory 1996].The second half discusses the existence problem of compact Clifford–Klein forms of pseudo-Riemannian homogeneous spaces, for which many new methods from different areas have been recently employed. We examine these various approaches in some typical cases. We also point out that Zimmer's examples on SL(n)/SL(m) [J. Amer. Math. Soc. 1994] and Shalom's examples on SL(n)/SL(2) [Ann. Math. 2000] are readily obtained as special cases of Kobayashi's criterion [Duke Math. J. 1992], where the former uses ergodic theory and restrictions of unitary representations, respectively, while the latter uses cohomologies of discrete groups.The article also explains some open problems and conjectures.