On the Residual Finiteness of Outer Automorphisms of Hyperbolic Groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable hyperbolic group is residually finite. As a result, we are able to prove that the group of outer automorphisms of every finitely generated Fuchsian group and of every free-by-finite group is residually finite.     

Malcev presentations for subsemigroups of direct products of coherent groups

The direct product of a free group and a polycyclic group is known to be coherent. This paper shows that every finitely generated subsemigroup of the direct product of a virtually free group and an abelian group admits a finite Malcev presentation. (A Malcev presentation is a presentation of a special type for a semigroup that embeds into a group. A group is virtually free if it contains a free subgroup of finite index.) By considering the direct product of two free semigroups, it is also shown that polycyclic groups, unlike nilpotent groups, can contain finitely generated subsemigroups that do not admit finite Malcev presentations.     

On the residual finiteness of outer automorphisms of relatively hyperbolic groups

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit group is residually finite.     

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.     

Galois groups of quantum group actions and regularity of fixed-point algebras

It is shown that, for a minimal and integrable action of a locally compact quantum group on a factor, the group of automorphisms of the factor leaving the fixed-point algebra pointwise invariant is identified with the intrinsic group of the dual quantum group. It is proven also that, for such an action, the regularity of the fixed-point algebra is equivalent to the cocommutativity of the quantum group.

     

Representation of the group of holomorphic symmetries of a real germ in the symmetry group of its model surface

Local polynomial models of real submanifolds of complex spaces were constructed and studied in a series of papers. Among the main features of model surfaces, there is the property that the dimension of the local group of holomorphic symmetries of a germ does not exceed that of the same group of the tangent model surface of this germ. In the paper, this assertion is rendered much stronger; namely, it is proved that the connected component of the identity element in the symmetry group of a nondegenerate germ is isomorphic as a Lie group to a subgroup of the symmetry group of its tangent model surface.     

The power graph of a finite group

The power graph of a group is the graph whose vertex set is the group, two elements being adjacent if one is a power of the other. We observe that non-isomorphic finite groups may have isomorphic power graphs, but that finite abelian groups with isomorphic power graphs must be isomorphic. We conjecture that two finite groups with isomorphic power graphs have the same number of elements of each order. We also show that the only finite group whose automorphism group is the same as that of its power graph is the Klein group of order 4.     

Birds of Moderately Different Feathers: Bandwagon Dynamics and the Threshold Heterogeneity of Network Neighbors

We present a simple combinatorial model of group decision‐making in a dichotomous choice situation. The independent variables in the model are group size and mean judgmental competence of group members. The dependent variables are the probabilities that the majority judgment of the group will be correct and that the best member of the group will be more likely to be correct than the group majority.

Two applications of this model are given: one a determination of isocompetence curves of groups of differing sizes and mean competence levels, the other an explication of the potential mechanisms accounting for Parkinson's only partly tongue‐in‐cheek observation that the point of ineffectiveness in a group seems to be reached when its total membership exceeds 20 or 21.     

Orthogonality of cosets relative to irreducible characters of finite groups

Studied is an assumption on a group that ensures that no matter how the group is embedded in a symmetric group, the corresponding symmetrized tensor space has an orthogonal basis of standard (decomposable) symmetrized tensors.     

关于群类的Sylow性质

群类理论是在有限可解群研究工作的基础上发展起来的,但近年来对有限群论的许多方面都起到越来越大的作用.在考察群类性质时,注意到一个Fitting类(?)在可解群G中的(?)内射子具有Sylow子群所具有的某些性质,并且关于Sylow定理中(Sy13),证明了对于群的本原子群,成立更强的结论.     

A class of symmetric and a class of Wiener group algebras

It is shown that connected groups of polynomial growth and compact extensions of nilpotent group have symmetric group algebras and that the group algebras of discrete solvable groups have the Wiener property.     

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     

The Generalised Pro-Fitting Subgroup of a Profinite Group

The generalised Fitting subgroup of a finite group is the group generated by all subnormal subgroups that are either nilpotent or quasisimple. The importance of this subgroup in finite group theory stems from the fact that it always contains its own centraliser, so that any finite group is an abelian extension of a group of automorphisms of its generalised Fitting subgroup. We define a class of profinite groups which generalises this phenomenon, and explore some consequences for the structure of profinite groups.     

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.     

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.     

On the nilpotency class of a multiplicative group of a modular group algebra of a dihedral 2-group

It is proved that the wreath product of a second-order group and the commutant of a dihedral group is imbedded into a multiplicative group of a modular group algebra of a dihedral group of order 2 ⁿ . This implies that the nilpotency class of the multiplicative group is equal to 2 ^n–2, i.e., to the order of the commutant of the dihedral group.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 1, pp. 39–45, January, 1995.     

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．     

Zero entropy subgroups of mapping class groups

Given a group action on a surface with a finite invariant set we investigate how the algebraic properties of the induced group of permutations of that set affects the dynamical properties of the group. Our main result shows that in many circumstances if the induced permutation group is not solvable then among the homeomorphisms in the group there must be one with a pseudo-Anosov component. We formulate this in terms of the mapping class group relative to the finite set and show the stronger result that in many circumstances (e.g. if the surface has boundary) if this mapping class group has no elements with pseudo-Anosov components then it is itself solvable.     

Automorphism group and dimension of ordered sets

It is shown that a finite groupG is isomorphic to the automorphism group of a two-dimensional ordered set if and only if it is a generalized wreath product of symmetric groups over an ordered index set that is a dual tree. Furthermore, every finite abelian group is isomorphic to the full automorphism group of a three-dimensional ordered set. Also every finite group is isomorphic to the automorphism group of an ordered set that does not contain an induced crown with more than four elements.     

The topology of group extensions of C systems

The paper is concerned with the topological and metric properties of group extensions of C systems. The basic theorem describes the topologically transitive component, the ergodic component, and the K component of a group extension of a C system. It is shown that each of these components is a group sub-bundle of a principal bundle in which the group extension acts. The frame flow on a manifold of negative curvature is seen to be a special case of a group of extension of a C system. It is shown that the space of frames on a compact three-dimensional manifold with negative curvature does not have any group sub-bundles, so that the frame flow on manifolds of this class is topologically transitive, ergodic, and a K system.