弱ss-置换子群与有限群的p-幂零性

称群G的一个子群H在G中弱ss-置换的,如果存在G的一个次正规子群T和包含在H中的G的一个ss-置换子群Hss使得G=HT且H∩T≤Hss．利用子群的弱ss-置换性得到了有限p-幂零群的一些新的刻画．     

The generic character table of a Sylow p-subgroup of a finite Chevalley group of type D4

Let U be a Sylow p-subgroup of the finite Chevalley group of type D₄ over the field of q elements, where q is a power of a prime p. We describe a construction of the generic character table of U.     

非极大交换子群皆正规的有限群

设H是有限群G的一个交换子群.如果H在G中的中心化子正是它本身,则称H为G的极大交换子群.本文主要研究每一非极大交换子群都正规的有限群的结构,对这类有限群给出其完全分类.     

Upper Bounds on Character Sums with Rational Function Entries

We obtain formulac and estimates for character sums of the type S(x,f,p^m)=∑x=1^p^mx(f(f)),where p^m is a prime power with m≥2,x is a multiplicative character(mod p^m),and f=f1/f2 is a rational function over Z.In particular,if p is odd,d=deg(f1) deg(f2) and d^*=max(deg(f1),deg(f2)) then we obtain |S(x,f,p^m)|≤(d-1)p^m(1-1/d^*) for any non-constant f (mod p) and primitive character x.For p=2 an extra factor of 2√2 is needed.     

有限交换环上极限线性群的Sylowp—子群

本文给出了有限交换局部环Ｒ上无限线性群ＧＬ（Ｒ）＝∪ｎＧＬｎＲ的Ｓｙｌｏｗｐ－子群的形式．令Ｍ是有限交换局部环Ｒ的唯一极大理想，ｋ＝Ｒ／Ｍ为Ｒ的剩余类域．用Ｘ（ｋ）表示ｋ的特征，并假定Ｐ与ｘ（ｋ）互素．作者证明了：ＧＬ（Ｒ）的任一Ｓｙｌｏｗｐ－子群Ｓ或者同构于的可数无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ≠２或Ｐ＝２，Ｘ（ｋ）β≡１（ｍｏｄ４））或者同构于Ｐｉ的无限直积与Ｐ（ｊ）的无限直积的直积（当Ｐ＝２，Ｘ（ｋ）β≡３（ｍｏｄ４）），这里，只是ＧＬ（ｅｐｉ）Ｒ（分别地，ＧＬ（２ｒｉ）Ｒ）的Ｓｙｌｏｗｐ－子群，Ｐ（ｊ））同构于Ｐ＝∪ｉ∈Ｉｐｉ，Ｉ是可数集．     

次正规子群与有限群的超可解性

通过Sylow子群的极大子群和次正规性,利用极小阶反例的方法,得出群p-幂零性和超可解性的结论.本文的创新改进之处在于结合Sylow子群的极大子群和次正规性,研究p-幂零性和超可解性的相关结论.     

High order moments of character sums

We establish the upper bound

with a prime and any positive integer, the sum being over all nonprincipal multiplicative characters .

     

Small solutions of the congruencea1x12 +a2x22 +a3x32 +a4x42 ≡c(modp)

For any integersa ₁,a ₂,a ₃,a ₄ andc witha ₁ a ₂ a ₃ a ₄≢0(modp), this paper shows that there exists a solutionX=(x ₁,x ₂,x ₃,x ₄) ∈Z ⁴ of the congruencea ₁ x ₁ ² +a ₂ x ₂ ² +a ₃ x ₃ ² +a ₄ x ₄ ² ≡c(modp) such that

On pseudo-random subsets of $${\mathbb Z _n}$$

The notion of pseudo-randomness of subsets of \({\mathbb Z_n}\) is defined, and the measures of pseudo-randomness are introduced. Then a construction (based on the use of hybrid character sums) will be presented for subsets of \({\mathbb Z_p}\) with strong pseudo-random properties.     

Lattice points in rational ellipsoids

We combine exponential sums, character sums and Fourier coefficients of automorphic forms to improve the best known upper bound for the lattice error term associated to rational ellipsoids.     

On gaps between quadratic non-residues in the Euclidean and Hamming metrics

The authors have recently introduced and studied a modification of the classical number theoretic question about the largest gap between consecutive quadratic non-residues and primitive roots modulo a prime p$p$, where the distances are measured in the Hamming metric on binary representations of integers. Here we continue to study the distribution of such gaps. In particular we prove the upper bound

_?p≤(0.117198…+o(1))logp/log2${?}_p\le (0.117198\dots +o\left(1\right))logp/log2$

for the smallest Hamming weight _?p${?}_p$ among prime quadratic non-residues modulo a sufficiently large prime p$p$. The Burgess bound on the least quadratic non-residue only gives _?p≤(0.15163…+o(1))logp/log2${?}_p\le (0.15163\dots +o\left(1\right))logp/log2$.     

A hybrid mean value formula involving Kloosterman sums and Hurwitz zeta-function

The main purpose of this paper is using the mean value theorem of Dirichlet L-function and the estimates for character sums to study the asymptotic properties of a hybrid mean value of Kloosterman sums with the weight of Hurwitz zeta-function and the Cochrane sums, and give an interesting mean value formula for it.     

Algebraic Cayley graphs over finite fields

A new algebraic Cayley graph is constructed using finite fields. It provides a more flexible source of expander graphs. Its connectedness, the number of connected components, and diameter bound are studied via Weil's estimate for character sums. Furthermore, we study the algorithmic problem of computing the number of connected components and establish a link to the integer factorization problem.