线性方程组解陪集

从不变子群的陪集的角度研究线性方程组的解陪集之间的关系,并找到了商空间的基与维数.     

Die Kristallpackung von (PPh4)2[NiCl4] · 2 MeCN und PPh4[CoCl0,6Br2,4(NCMe)]

The Crystal Packings of (PPh₄)₂[NiCl₄] · 2 MeCN and PPh₄[CoCl_0.6Br_2.4(NCMe)] (PPh₄)₂[NiCl₄] · 2 MeCN was obtained from the reaction of PPh₄Cl and NiCl₂ in acetonitrile in the presence of S₂Cl₂, PPh₄[Cl₂H] being a side product. The product of the reaction of CoS₂ with S₂Br₂ (containing rests of S₂Cl₂) at 400 °C was treated with PPh₄Br in acetonitrile yielding PPh₄Br₃ and PPh₄[CoCl_0.6Br_2.4(NCMe)]. The crystal structures of the title compounds were determined by X‐ray diffraction. (PPh₄)₂[NiCl₄] · 2 MeCN (space group I 4, a = 1839.3 pm, c = 1375.3 pm) has a crystal packing derived from the BiPh₄[ClO₄] structure type with a fourfold increased unit cell and one half of the ClO₄^– positions substituted by pairsof acetonitrile molecules. The crystal structure of PPh₄[CoCl_0.6Br_2.4(NCMe)] (space group I4₁/a, a = 1804.7 pm, c = 3198.8 pm) is related to the AsPh₄[RuNCl₄] type with an eightfold increased unit cell. The [CoCl_0.6Br_2.4(NCMe)]^– ions are disordered in two orientations and some halogen positions are randomly occupied by Cl and Br atoms. Family trees of group–subgroup relations show the symmetry relations.     

Geodesics as products of one-parameter subgroups in compact lie groups and homogeneous spaces

We study the geodesic equation for compact Lie groups G and homogeneous spaces $G/H$, and we prove that the geodesics are orbits of products $\exp (t{X}_1)\cdots \exp (t{X}_N)$ of one-parameter subgroups of G, provided that a simple algebraic condition for the Riemannian metric is satisfied. For the group $SO(3)$, we relate this type of geodesics to the free motion of a symmetric top. Moreover, by using series of Lie subgroups of G, we construct a wealth of metrics having the aforementioned type of geodesics.     

The rational cuspidal subgroup of J0(p2M)$J_0(p^2M)$ with M squarefree

For a positive integer N, let $X_0(N)$ be the modular curve over $\mathbf{Q}$ and $J_0(N)$ its Jacobian variety. We prove that the rational cuspidal subgroup of $J_0(N)$ is equal to the rational cuspidal divisor class group of $X_0(N)$ when $N=p^{2}M$ for any prime p and any squarefree integer M. To achieve this, we show that all modular units on $X_0(N)$ can be written as products of certain functions $F_{m,h}$, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on $X_0(N)$ under a mild assumption.     

((∈),(∈)V(q)(λ,μ))-模糊正规子群的若干性质

本文研究了((∈),(∈)V(q)(λ,μ))-模糊正规子群的一些新的性质.利用反扩张原理获得了它的满同态像与同态原像的相关结果,丰富了文献[1]中有关((∈),(∈)V(q)(λ,μ))-模糊正规子群的研究结果.     

有限群的素数幂阶S-拟正规嵌入子群

设G为有限p-可解群,其中p为|G|的奇素因子.若P为G的Sylow p-子群且最小生成系含d个元素.考虑集合M_d(P)={P_1,…,P_d},其中P_1,…,P_d是P的极大子群且满足(?)P_i=φ(P).证明了若M_d(P)中每个元在G中是S-拟正规嵌入的,则G为p-超可解群.作为应用,还得到了一些进一步的结论.