On the order of elements in long minimal zero-sum sequences

Let G be a finite abelian group and S = a minimal zero-sum sequence in G of maximal length |S| = l. We study the order of the elements      

On zero-sum sequences of prescribed length

Summary. Let k ≥ 1 be any integer. Let G be a finite abelian group of exponent n. Let s_k(G) be the smallest positive integer t such that every sequence S in G of length at least t has a zero-sum subsequence of length kn. We study this constant for groups when d = 3 or 4. In particular, we prove, as a main result, that for every k ≥ 4, and for every prime p ≥ 5.     

A result on the sum of element orders of a finite group

Let G be a finite group and $$\psi (G)=\sum _{g\in {G}}{o(g)}$$. There are some results about the relation between $$\psi (G)$$ and the structure of G. For instance, it is proved that if G is a group of order n and $$\psi (G)>\dfrac{211}{1617}\psi (C_n)$$, then G is solvable. Herzog et al. in (J Algebra 511:215–226, 2018) put forward the following conjecture: Conjecture. If G is a non-solvable group of order n, then $$\begin{aligned} {\psi (G)}\,{\le }\,{{\dfrac{211}{1617}}{\psi (C_n)}}, \end{aligned}$$with equality if and only if $$G \cong A_5$$. In particular, this inequality holds for all non-Abelian simple groups. In this paper, we prove a modified version of Herzog’s Conjecture.     

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.     

Orders generated by character values

Let $$K:=\mathbb {Q}(G)$$ be the number field generated by the complex character values of a finite group G. Let $$\mathbb {Z}_K$$ be the ring of integers of K. In this paper we investigate the suborder $$\mathbb {Z}[G]$$ of $$\mathbb {Z}_K$$ generated by the character values of G. We prove that every prime divisor of the order of the finite abelian group $$\mathbb {Z}_K/\mathbb {Z}[G]$$ divides |G|. Moreover, if G is nilpotent, we show that the exponent of $$\mathbb {Z}_K/\mathbb {Z}[G]$$ is a proper divisor of |G| unless $$G=1$$. We conjecture that this holds for arbitrary finite groups G.     

On the Second Smallest and the Largest Normalized Laplacian Eigenvalues of a Graph

Acta Mathematicae Applicatae Sinica, English Series - Let G be a simple connected graph with order n. Let $$\mathcal{L}(G)$$ and $$\mathcal{Q}(G)$$ be the normalized Laplacian and normalized...     

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...     

The Growth of Composite Meromorphic Functions

Let f be a transcendental meromorphic function of order $ \rho _f $ , g be a transcendental entire function of lower order $\lambda _g (\lambda _g \lt + \infty ) $ with $ \sum _{a\not = \infty }\delta (a,g)= 1 $ , then $$\overline {\mathop {{\rm lim}}\limits_{r \to \infty } } \log {{\left( {T\left( {r,f\left( g \right)} \right)} \right)} \mathord{\left/{\vphantom {{\left( {T\left( {r,f\left( g \right)} \right)} \right)} {T\left( {r,g} \right)}}} \right. \kern-\nulldelimiterspace} {T\left( {r,g} \right)}} = \pi \rho f.$$     

The domination game played on diameter 2 graphs

Aequationes mathematicae - Let $$\gamma _g(G)$$ be the game domination number of a graph G. It is proved that if $$\mathrm{diam}(G) = 2$$ , then $$\gamma _g(G) \le \left\lceil \frac{n(G)}{2}...     

Finite Groups with Formation Subnormal Normalizers of Sylow Subgroups

Mathematical Notes - Let $$\mathfrak{F}$$ be a formation. Properties of the class $$\mathrm{w}^{*}\mathfrak{F}$$ of all groups $$G$$ for which $$\pi(G)\subseteq\pi(\mathfrak{F})$$ and the...     

Fundamental groups of moduli of principal bundles on curves

Geometriae Dedicata - Let X be a compact connected Riemann surface of genus g, with $$g\, \ge \, 2$$ , and let $${\text {G}}$$ be a connected semisimple affine algebraic group defined over...     

Exponent of a finite group of odd order with an involutory automorphism

Let $$w = w(x_1, \ldots , x_n)$$ be a non-trivial word of n-variables. The word map on a group G that corresponds to w is the map $$\widetilde{w}: G^n\rightarrow G$$ where $$\widetilde{w}((g_1, \ldots , g_n)) := w(g_1, \ldots , g_n)$$ for every sequence $$(g_1, \ldots , g_n)$$ . Let $$\mathcal G$$ be a simple and simply connected group which is defined and split over an infinite field K and let $$G =\mathcal G(K)$$ . For the case when $$w = w_1w_2 w_3 w_4$$ and $$w_1, w_2, w_3, w_4$$ are non-trivial words with independent variables, it has been proved by Hui et al. (Israel J Math 210:81–100, 2015) that $$G{\setminus } Z(G) \subset {{\text { Im}}}\,\widetilde{w}$$ where Z(G) is the center of the group G and $${{\text { Im}}}\, {\widetilde{w}}$$ is the image of the word map $$\widetilde{w}$$ . For the case when $$G = {{\text {SL}}}_n(K)$$ and $$n \ge 3$$ , in the same paper of Hui et al. (2015) it was shown that the inclusion $$G{\setminus } Z(G)\subset {{\text { Im}}}\,\widetilde{w}$$ holds for a product $$w = w_1w_2 w_3$$ of any three non-trivial words $$ w_1, w_2, w_3$$ with independent variables. Here we extent the latter result for every simple and simply connected group which is defined and split over an infinite field K except the groups of types $$B_2, G_2$$ .     

Volume bounds of the Ricci flow on closed manifolds

Mathematische Zeitschrift - Let $$\{g(t)\}_{t\in [0,T)}$$ be the solution of the Ricci flow on a closed Riemannian manifold $$M^n$$ with $$n\ge 3$$ . Without any assumption, we derive lower volume...     

Finite simple groups whose Sylow 2-subgroups possess an extra-special subgroup of index 2

Let G be a finite simple group with Sylow 2-subgroup T. If there is an extra-special sub-group of index 2 in T, then G is isomorphic to one of the following groups: $$\mathop {M_{11,} }\limits_{U_3 (q),}$$ for an appropriate odd q.     

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| =...     

A Tree-valued Markov Process Associated with an Admissible Family of Branching Mechanisms

Let E?R be an interval. By studying an admissible family of branching mechanisms{ψt,t ∈E} introduced in Li [Ann. Probab., 42, 41-79(2014)], we construct a decreasing Levy-CRT-valued process {Tt, t ∈ E} by pruning Lévy trees accordingly such that for each t ∈E, Tt is a ψt-Lévy tree. We also obtain an analogous process {Tt*,t ∈E} by pruning a critical Levy tree conditioned to be infinite. Under a regular condition on the admissible family of branching mechanisms, we show that the law of {Tt,t ∈E} at the ascension time A := inf{t ∈E;Tt is finite} can be represented by{Tt*,t∈E}.The results generalize those studied in Abraham and Delmas [Ann. Probab., 40, 1167-1211(2012)].     

Sign changes in the prime number theorem

The Ramanujan Journal - Let V(T) denote the number of sign changes in $$\psi (x) - x$$ for $$x\in [1, T]$$ . We show that $$\liminf _{T\rightarrow \infty } V(T)/\log T\ge \gamma _{1}/\pi +...     

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.     

Semisimplicity of Some Class of Operator Algebras on Banach Space

Let G be a locally compact abelian group and let be a representation of G by means of isometries on a Banach space. We define W_T as the closure with respect to the weak operator topology of the set where is the Fourier transform of f ∈L¹(G) with respect to the group T. Then W_T is a commutative Banach algebra. In this paper we study semisimlicity problem for such algebras. The main result is that if the Arveson spectrum sp(T) of T is scattered (i.e. it does not contain a nonempty perfect subset) then the algebra W_T is semisimple. Some related problems are also discussed.     

Roots of crosscap slides and crosscap transpositions

Periodica Mathematica Hungarica - Let $$N_{g}$$ denote a closed nonorientable surface of genus g. For $$g \ge 2$$ the mapping class group $$\mathcal {M}(N_{g})$$ is generated by Dehn twists and one...