A complex for right-angled Coxeter groups

We associate to each right-angled Coxeter group a 2-dimensional complex. Using this complex, we show that if the presentation graph of the group is planar, then the group has a subgroup of finite index which is a 3-manifold group (that is, the group is virtually a 3-manifold group). We also give an example of a right-angled Coxeter group which is not virtually a 3-manifold group.

     

The permutation class group of a finite group

Analogously to the projective class group, the permutation class group of a finite group π can be defined as the group of equivalence classes of direct summands of integral permutation modules modulo permutation modules. It is shown that this group behaves nicely with respect to localization and completion, which then is used to prove that contrary to the projective class group - it is not always a torsion group. More precisely, the rank of the permutation class of group is computed.     

The structure of abelian pro-Lie groups

A pro-Lie group is a projective limit of a projective system of finite dimensional Lie groups. A prodiscrete group is a complete abelian topological group in which the open normal subgroups form a basis of the filter of identity neighborhoods. It is shown here that an abelian pro-Lie group is a product of (in general infinitely many) copies of the additive topological group of reals and of an abelian pro-Lie group of a special type; this last factor has a compact connected component, and a characteristic closed subgroup which is a union of all compact subgroups; the factor group modulo this subgroup is pro-discrete and free of nonsingleton compact subgroups. Accordingly, a connected abelian pro-Lie group is a product of a family of copies of the reals and a compact connected abelian group. A topological group is called compactly generated if it is algebraically generated by a compact subset, and a group is called almost connected if the factor group modulo its identity component is compact. It is further shown that a compactly generated abelian pro-Lie group has a characteristic almost connected locally compact subgroup which is a product of a finite number of copies of the reals and a compact abelian group such that the factor group modulo this characteristic subgroup is a compactly generated prodiscrete group without nontrivial compact subgroups.Mathematics Subject Classification (1991): 22B, 22E     

Hereditary crossed products

We characterize when a crossed product order over a maximal order in a central simple algebra by a finite group is hereditary. We need only concentrate on the cases when the group acts as inner automorphisms and when the group acts as outer automorphisms. When the group acts as inner automorphisms, the classical group algebra result holds for crossed products as well; that is, the crossed product is hereditary if and only if the order of the group is a unit in the ring. When the group is acting as outer automorphisms, every crossed product order is hereditary, regardless of whether the order of the group is a unit in the ring.

     

Monomorphism categories associated with symmetric groups and parity in finite groups

Monomorphism categories of the symmetric and alternating groups are studied via Cayley’s Em-bedding Theorem. It is shown that the parity is well defined in such categories. As an application, the parity in a finite group G is classified. It is proved that any element in a group of odd order is always even and such a group can be embedded into some alternating group instead of some symmetric group in the Cayley’s theorem. It is also proved that the parity in an abelian group of even order is always balanced and the parity in an nonabelian group is independent of its order.     

Coorbits of Quantum Homogeneous Spaces

The notion of coorbits for spaces with quantum group actions is introduced. A space with a quantum group action is given by a pair of algebras: an associative algebra which is the analog of a classical topological space, and a Hopf algebra which is the analog of a classical topological group. The Hopf algebra acts on the associative algebra via a comodule structure mapping which is also an algebra homomorphism. For a space with a quantum group action, a coorbit is a pair of spaces given by the image and the kernel of an algebra homomorphism from the associative algebra to the Hopf algebra. The coorbits of several types of quantum homogeneous spaces are discussed. In the case when the associative algebra is the group algebra of a group and the Hopf algebra is a quotient of the group algebra, the connection between the set of coorbits and the character group is established.     

Coleman自同构群的投射极限

在这篇注记中,利用群的投射极限性质给出了有限可解群的Coleman自同构群的一个具体构造.作为应用,证明了二面体群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

Relative hyperbolicity and similar properties of one-generator one-relator relative presentations with powered unimodular relator

A group obtained from a nontrivial group by adding one generator and one relator which is a proper power of a word in which the exponent sum of the additional generator is one contains the free square of the initial group and almost always (with one obvious exception) contains a non-abelian free subgroup. If the initial group is involution-free or the relator is at least third power, then the obtained group is SQ-universal and relatively hyperbolic with respect to the initial group.     

Group signature schemes with forward secure properties

A group signature scheme allows a group member to sign messages anonymously on behalf of the group. However, in the case of a dispute, the identity of a signature can be revealed by a designated entity. We introduce a forward secure schemes into group signature schemes. When the group public key remains fixed, a group signing key evolves over time. Because the signing key of a group member is evolving at time, the possibility of the signing key being exposed is decreased. We propose a forward secure group signature scheme based on Ateniese and Camenisch et al.’s group signature scheme. The security is analyzed and the comparisons between our scheme with other group signature schemes are made.     

Coleman自同构群的投射极限

在这篇注记中，利用群的投射极限性质给出了有限可解群的Coleman自同构群的一个具体构造.作为应用，证明了二面体群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

大型群体决策问题中的简单多数规则（3个评选对象场合）

一个群体决策问题取决于两个因素,一个是群体决策的规则,另一个是投票。当选定群体决策规则时,一个群体决策问题由投票完全决定,此时,群体决策问题与投票之间一一对应。简单多数规则是个简单且被广泛采用的群体决策规则,但它有缺陷,我们可举出些群体决策问题使用简单多数规则没法从投票得到最后群体决策的结果。这里我们将给出一个简单多数规则的有趣性质,即在3个评选对象场合,使用简单多数规则没法从投票得到最后群体决策结果的n个评选人的群体决策问题的个数与所有n个评选人的群体决策问题的个数之比当评选人个数n趋向无穷时趋于零,这说明3个评选对象的大型群体决策场合,简单多数规则的缺陷不严重。     

拟L商群的同态与同构

HX群是由一个群向其幂集上提升的群结构．文[1]将HX群推广到一类特殊的格-原子格L上,称之为AHX群．本文研究了一类特殊的AHX群-拟L商群的同态与同构,获得与拟商群相类似的结果．     

两类Cayley的向图的同构问题

证明了对m=1,2,3,有限广义双循环群B(Q8)是m-DCI群当且仅当它的极大交换子群L是m-DCI群且4|-1mm|L|;有限广义二面体群D是m-DCI群当且仅当它的极大交换子群K是m-DCI群且2|-1mm|K|.     

状态离散的确定性多阶段群体决策的满意策略

为了求解状态离散的确定性多阶段群体决策问题,建立了多阶段群体决策模型,定义了群体Pareto最优策略、群体满意策略等概念,依据Bellman最优性原理,提出了多阶段群体决策问题的逆向递推算法,并通过引入偏好关系,得到了各阶段的子过程群体满意策略以及全过程群体满意策略,最后给出了一个计算实例．     

Structure Properties and Real Continuous Bounded 2-Cohomology of Locally Compact Groups

With the help of some structure results for locally compact groups, the second real continuous bounded cohomology group of a connected locally compact group is described and it is proved that the corresponding group is finite-dimensional for any almost connected locally compact group.     

Hidden symmetries and nonlocal group generators for ordinary differential equations

Hidden symmetries of ordinary differential equations (ODEs)are studied with nonlocal group generators. General forms aregiven for an exponential nonlocal group generator of an ODEthat is reduced from a higher-order ODE, which is expressedin canonical variables and which is invariant under a two-parameterLie group. The nonlocal group generator identifies a type Ihidden symmetry. Type II hidden symmetries are found in somereduction pathways of an ODE invariant under a solvable, nonabelian,three-parameter Lie group. The algorithm for the appearanceof the type II hidden symmetry is stated. General forms forthe reduced nonlocal group generator, which identifies the typeII hidden symmetry, are presented when the other two commutingoriginal group generators are in normal form.     

On constructive nilpotent groups

We prove the following: (1) a torsion-free class 2 nilpotent group is constructivizable if and only if it is isomorphic to the extension of some constructive abelian group included in the center of the group by some constructive torsion-free abelian group and some recursive system of factors; (2) a constructivizable torsion-free class 2 nilpotent group whose commutant has finite rank is orderably constructivizable.     

关于群体决策的偏比映射

本文引进群体决策的决策个体和决策群体关于供选方案对的偏比度,以及偏比泛函概念.由此,定义了群体在供选方案集上的偏比映射,构造了相应的偏比公理系,并且建立了关于群体决策的偏比度分析的基本理论.最后,给出一个利用偏比映射进行群体排序的方法.特别是,文中表明,满足本文所提出的所有偏比公理的偏比映射是存在的.