关于一类中心循环的有限P-群的自同构群

确定了一类中心循环的有限p-群G的自同构群.设G=X_3(p~m)~(*n)*Z_(p~(m+r)),其中m≥1,n≥1和r≥0,并且X_3(p~m)=x,y|x~(p~m)=y~(p~m)=1,[x,y]~(p~m)=1,[x,[x,y]]=[y,[x,y]]=1.Aut_nG表示Aut G中平凡地作用在N上的元素形成的正规子群,其中G'≤N≤ζG,|N|=p~(m+s),0≤s≤r,则(i)如果p是一个奇素数,那么AutG/Aut_nG≌Z_(p~((m+s-1)(p-1))),Aut_nG/InnG≌Sp(2n,Z_(p~m))×Z_(p~(r-s)).(ii)如果p=2,那么AutG/Aut_nG≌H,其中H=1(当m+s=1时)或者Z_(2~(m+s-2))×Z_2(当m+s≥2时).进一步地,Aut_nG/InnG≌K×L,其中K=Sp(2n,Z_(2~m))(当r0时)或者O(2n,Z_(2~m))(当r=0时),L=Z_(2~(r-1))×Z_2(当m=1,s=0,r≥1时)或者Z_(2~(r-s)).     

Winter定理和Dietz定理的推广

设G是由中心扩张1→Z_p^m→G→Z_p×…Z_p所决定的有限p-群,且|G’|≤p.确定了G的自同构群结构,推广了Winter和Dietz的工作     

On m- $$\sigma$$ σ -embedded subgroups of finite groups

Acta Mathematica Hungarica - Let $$\sigma =\{\sigma_i |i\in I\}$$ be some partition of the set of all primes $$\mathbb{P}$$ and G be a finite group. A group is said to be $$\sigma$$ -primary if it...     

增益图与其底图的正惯性指数之间的关系

设$\mathbb{T}$是模为1的复数乘法子群.图$G=(V,E)$,这里$V,E$分别表示图的点和边.增益图是将底图中的每条边赋于$\mathbb{T}$中的某个数值$\varphi(v_iv_j)$,且满足$\varphi(v_iv_j) =\overline{\varphi(v_jv_i)}$.将赋值以后的增益图表示为$(G,\varphi)$.设$i_+(G,\varphi)$和$i_+(G)$分别表示增益图与底图的正惯性指数,本文证明了如下结论: $$ - c( G ) \le {i_ + } ( {G,\varphi } ) - {i_ + }( G ) \le c( G ), $$ 这里$c(G)$表示圈空间维数,并且刻画了等号成立时候的所有极图.     

On sums of polynomial-type exceptional units in $$\varvec{\mathbb {Z}}/\varvec{n\mathbb {Z}}$$Z/nZ

A unit u in a commutative ring with unity R is called exceptional if $$u-1$$ is also a unit. We introduce the notion of a polynomial version of this (abbreviated as $$f\hbox {-exunits}$$) for any $$f(X) \in \mathbb {Z}[X]$$. We find the number of representations of a non-zero element of $$\mathbb {Z}/n\mathbb {Z}$$ as a sum of two f-exunits for an infinite family of polynomials f of each degree $$\ge 1$$. We also derive the exact formulae for certain infinite families of linear and quadratic polynomials. This generalizes a result proved by Sander (J Number Theory 159:1–6, 2016).     

换位子群是不可分Abel群的有限秩可除幂零群

完整地确定了换位子群是不可分Abel群的有限秩可除幂零群的结构,证明了下面的定理.设G是有限秩的可除幂零群,则G的换位子群是不可分Abel群当且仅当G'=Q或Q_p/Z且G可以分解为G=S×D,其中当G'=Q时,■当G'=Q_p/Z时,S有中心积分解S=S_1*S_2*…*S_r,并且可以将S形式化地写成■其中■,式中s,t都是非负整数,Q是有理数加群,π_κ(k=1,2,…,t)是某些素数的集合,满足π_1■Cπ_2■…■π_t,Q_π_k={m/n|(m,n)=1,m∈Z,n为正的π_k-数}.进一步地,当G'=Q时,(r;s;π_1,π_2,…,π_t)是群G的同构不变量;当G'=Q_p/Z时,(p,r;s;π_1,π_2,…,πt)是群G的同构不变量.即若群H也是有限秩的可除幂零群,它的换位子群是不可分Abel群,那么G同构于H的充分必要条件是它们有相同的不变量.     

Sufficient conditions for local equivalence of mappings

Let $$f,g:({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^m,0)$$ be $$C^{r+1}$$ mappings and let $$Z=\{x\in \mathbf {\mathbb {R}}^n:\nu (df (x))=0\}$$ , $$0\in Z$$ , $$m\le n$$ . We will show that if there exist a neighbourhood U of $$0\in {\mathbb {R}}^n$$ and constants $$C,C'>0$$ and $$k>1$$ such that for $$x\in U$$ $$\begin{aligned}&\nu (df(x))\ge C{\text {dist}}(x,Z)^{k-1}, \\&\left| \partial ^{s} (f_i-g_i)(x) \right| \le C'\nu (df(x))^{r+k-|s|}, \end{aligned}$$ for any $$i\in \{1,\dots , m\}$$ and for any $$s \in \mathbf {\mathbb {N}}^n_0$$ such that $$|s|\le r$$ , then there exists a $$C^r$$ diffeomorphism $$\varphi :({\mathbb {R}}^n,0)\rightarrow ({\mathbb {R}}^n,0)$$ such that $$f=g\circ \varphi $$ in a neighbourhood of $$0\in {\mathbb {R}}^n$$ . By $$\nu (df)$$ we denote the Rabier function.     

Iterated sumsets and setpartitions

The Ramanujan Journal - Let $$G\cong {\mathbb {Z}}/m_1{\mathbb {Z}}\times \cdots \times {\mathbb {Z}}/m_r{\mathbb {Z}}$$ be a finite abelian group with $$m_1\mid \cdots \mid m_r=\exp (G)$$ . The...     

On dualization of a result of Bryce and Cossey theory

Acta Mathematica Hungarica - Let $$\sigma$$ be a partition of the set of all primes $$\mathbb{P}$$ . Let G be a finite group and $$\mathfrak{F}$$ be a Fitting class of finite groups. In the theory...     

On the average sum of the kth divisor function over values of quadratic polynomials

The Ramanujan Journal - Let $$F(x) \in \mathbb {Z}[x_1 , x_2 ,\ldots , x_n ]$$ , $$n\ge 3$$ , be an n-variable quadratic polynomial with a nonsingular quadratic part. Using the circle method we...     

On the average number of subgroups of the group $${\mathbb {Z}}_m \times {\mathbb {Z}}_n$$ Z m × Z n with k-th power sequence

The Ramanujan Journal - Let $$\mathbb {Z}_{n}$$ be the additive group of residue classes modulo n. Let s(m, n) denote the total number of subgroups of the group $$\mathbb {Z}_{m} \times...     

The decomposition of permutation module for infinite Chevalley groups

Let G be a connected reductive group defined over F_q, the finite field with q elements. Let B be a Borel subgroup defined over F_q. In this paper, we completely determine the composition factors of the induced module M(tr) = kG ■tr(where tr is the trivial B-module) for any field k.     

On $ {p} $‐groups of maximal class

Let R be the ring $ {\mathbb Z}[x]/\left({{x^p-1}\over{x-1}}\right) = {\mathbb Z}[\bar{x}] $ and let $ \mathfrak {a} $ be the ideal of R generated by $ (\bar{x}-1) $ . In this paper, we discuss the structure of the $ {\mathbb Z}[C_p] $‐module $ (R/\mathfrak {a}^{n-1}) \wedge (R/\mathfrak {a}^{n-1}) $, which plays an important role in the theory of p‐groups of maximal class (see 2 - 5 ). The generators of this module allow us to obtain the defining relations of some important examples of p‐groups of maximal class with Y₁ of class two. In particular we obtain the best possible estimates for the degree of commutativity of p‐groups of maximal class with Y₁ of class two. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

The Navarro–Tiep Galois conjecture for $$p=2$$ p = 2

We investigate the upper $$FC-$$ central series of the unit group of an integral group ring $${\mathbb Z}G$$ of a periodic group G. We prove that $${\mathcal U}={{\mathcal U}}_1({\mathbb Z}G)$$ has $$FC-$$ central height one if and only if the $$FC-$$ hypercenter of $${{\mathcal U}}_1({\mathbb Z}G)$$ is contained in the normalizer of the trivial units. Further, in these conditions, the $$FC-$$ hypercenter of the unit group is non-central if and only if G is a $$Q^{*}-$$ group. Let $$H \vartriangleleft {\mathcal U}, H$$ contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of $${{\mathcal U}}_1({\mathbb Z}G)$$ .     

Quadratic polynomials represented by norm forms

Let ${P(t) \in \mathbb{Q}[t]}$ be an irreducible quadratic polynomial and suppose that K is a quartic extension of ${\mathbb{Q}}$ containing the roots of P(t). Let ${{\bf N}_{K/\mathbb{Q}}({\rm x})}$ be a full norm form for the extension ${K/\mathbb{Q}}$ . We show that the variety $$\begin{array}{ll}P(t)={\bf N}_{K/\mathbb{Q}}({\rm x})\neq 0\end{array}$$ satisfies the Hasse principle and weak approximation. The proof uses analytic methods.     

A Characterization of T2g+1,2 among Alternating Knots

Acta Mathematica Sinica, English Series - Let K be a genus g alternating knot with Alexander polynomial $${\Delta _K}(T) = \sum\nolimits_{i = - g}^g {{a_i}{T^i}}$$ . We show that if |ag| =...     

Normes cyclotomiques naïves et unités logarithmiques

We compute the \({\mathbb {Z}}\)-rank of the subgroup \(\widetilde{E}_K =\bigcap _{n\in {\mathbb {N}}} N_{K_n/K}(K_n^\times )\) of elements of the multiplicative group of a number field K that are norms from every finite level of the cyclotomic \({\mathbb {Z}}_\ell \)-extension \(K^c\) of K. Thus we compare its \(\ell \)-adification \({\mathbb {Z}}_\ell \otimes _{\mathbb {Z}}\widetilde{E}_K\) with the group of logarithmic units \(\widetilde{\varepsilon }_K\). By the way we point out an easy proof of the Gross–Kuz’min conjecture for \(\ell \)-undecomposed extensions of abelian fields.     

Fractional Gaussian estimates and holomorphy of semigroups

Let $$\Omega \subset {\mathbb {R}}^N$$ be an arbitrary open set, $$0<s<1$$ and denote by $$(e^{-t(-\Delta )_{{{\mathbb {R}}}^N}^s})_{t\ge 0}$$ the semigroup on $$L^2({{\mathbb {R}}}^N)$$ generated by the fractional Laplace operator. In the first part of the paper, we show that if T is a self-adjoint semigroup on $$L^2(\Omega )$$ satisfying a fractional Gaussian estimate in the sense that $$|T(t)f|\le Me^{-bt(-\Delta )_{{{\mathbb {R}}}^N}^s}|f|$$, $$0\le t \le 1$$, $$f\in L^2(\Omega )$$, for some constants $$M\ge 1$$ and $$b\ge 0$$, then T defines a bounded holomorphic semigroup of angle $$\frac{\pi }{2}$$ that interpolates on $$L^p(\Omega )$$, $$1\le p<\infty $$. Using a duality argument, we prove that the same result also holds on the space of continuous functions. In the second part, we apply the above results to the realization of fractional order operators with the exterior Dirichlet conditions.     

循环群$Z_{2}$的模向量不变式

Let G be the finite cyclic group Z_2 and V be a vector space of dimension 2n with basis x_1,...,x_n,y_1,...,y_n over the field F with characteristic 2.If σ denotes a generator of G,we may assume that σ（x_i）= ayi,σ（y_i）= a～-1x_i,where a ∈ F.In this paper,we describe the explicit generator of the ring of modular vector invariants of F[V]～G.We prove that F[V]～G = F[l_i = x_i ＋ ay_i,q_i = x_iy_i,1 ≤ i ≤ n,M_I = X_I ＋ a～-I-Y_I],where I∈An = {1,2,...,n},2 ≤-I-≤ n.