Countable Primitive Groups

We give a complete characterization of countable primitive groups in several settings including linear groups, subgroups of mapping class groups, groups acting minimally on trees and convergence groups. The latter category includes as a special case Kleinian groups as well as subgroups of word hyperbolic groups. As an application we calculate the Frattini subgroup in many of these settings, often generalizing results that were only known for finitely generated groups. In particular, we answer a question of G. Higman and B.H. Neumann on the Frattini group of an amalgamated product. Received: January 2006, Revision: May 2006, Accepted: May 2006     

Homomorphisms between A-Projective Abelian Groups and Left Kasch-Rings

Glaz and Wickless introduced the class G of mixed abelian groups A which have finite torsion-free rank and satisfy the following three properties: i) A _p is finite for all primes p, ii) A is isomorphic to a pure subgroup of _P A _P and iii) Hom(A, tA) is torsion. A ring R is a left Kasch ring if every proper right ideal of R has a non-zero left annihilator. We characterize the elements A of G such that E(A)/tE(A) is a left Kasch ring, and discuss related results.     

Homomorphisms from Automorphism Groups of Free Groups

The automorphism group of a finitely generated free group isthe normal closure of a single element of order 2. If m <n, then a homomorphism Aut(F_n)Aut(F_m) can have image of cardinalityat most 2. More generally, this is true of homomorphisms fromAut(F_n) to any group that does not contain an isomorphic imageof the symmetric group S_n+1. Strong restrictions are also obtainedon maps to groups that do not contain a copy of W_n = (Z/2)ⁿ S_n, or of Z^n–1. These results place constraints on howAut(F_n) can act. For example, if n 3, any action of Aut(F_n)on the circle (by homeomorphisms) factors through det : Aut(F_n)Z₂.2000 Mathematics Subject Classification 20F65, 20F28 (primary).     

模糊群的模糊同态

引进模糊群的模糊同态概念，给出模糊同态下子模糊群(正规子模糊群)间的对应关系’建立模糊群的模糊同态基本定理，同时讨论模糊群的若干性质，得到一系列等价条件。     

On Homomorphisms of Fuzzy Groups

Clarification is suggested for the early notion of homomorphisms of fuzzy groups. Conditions are considered for fulfilment of various properties of homomorphisms of ordinary groups as well as properties specific to systems with a fuzzy operation. Correctness is discussed of the introduced notion of preservation of a fuzzy operation.     

On Homomorphisms of Fuzzy Groups

Clarification is suggested for the early notion of homomorphisms of fuzzy groups. Conditions are considered for fulfilment of various properties of homomorphisms of ordinary groups as well as properties specific to systems with a fuzzy operation. Correctness is discussed of the introduced notion of preservation of a fuzzy operation.     

Abstract Homomorphisms of Algebraic Groups

In this paper we are concerned with abstract homomorphisms fromone algebraic group to another. That is, group homomorphismswhere there is no assumption of rationality. Let K be an infiniteperfect field and L an algebraically closed field of the samecharacteristic. Throughout G(K) will denote a Chevalley groupof universal type over K.     

Cylinder Homomorphisms and Chow Groups

Let X be a projective algebraic manifold of dimension n (over C), CH¹(X) the Chow group of algebraic cycles of codimension l on X, modulo rational equivalence, and A¹(X) ? CH¹(X) the subgroup of cycles algebraically equivalent to zero. We say that A¹(X) is finite dimensional if there exists a (possibly reducible) smooth curve T and a cycle z∈CH¹(Γ × X) such that z_*:A¹(Γ)-A¹(X) is surjective. There is the well known Abel-Jacobi map λ₁:A¹(X)-J(X), where J(X) is the lth Lieberman Jacobian. It is easy to show that A¹(X)→J(X) A¹(X) finite dimensional. Now set with corresponding map A^*(X)→J(X). Also define Level . In a recent book by the author, there was stated the following conjecture: where it was also shown that (?) in (**) is a consequence of the General Hodge Conjecture (GHC). In this present paper, we prove A^*(X) finite dimensional ?? Level (H^*(X)) ≤ 1 for a special (albeit significant) class of smooth hypersurfaces. We make use of the family of k-planes on X, where ([…] = greatest integer function) and d = deg X; moreover the essential technical ingredients are the Lefschetz theorems for cohomology and an analogue for Chow groups of hypersurfaces. These ingredients in turn imply very special cases of the GHC for our choice of hypersurfaces X. Some applications to the Griffiths group, vanishing results, and (universal) algebraic representatives for certain Chow groups are given.     

Realizations of Countable Groups as Fundamental Groups of Compacta

It has been an open question for a long time whether every countable group can be realized as a fundamental group of a compact metric space. Such realizations are not hard to obtain for compact or metric spaces but the combination of both properties turn out to be quite restrictive for the fundamental group. The problem has been studied by many topologists (including Cannon and Conner) but the solution has not been found. In this paper we prove that any countable group can be realized as the fundamental group of a compact subspace of ${\mathbb{R}^4}$ . According to the theorem of Shelah [10] such space can not be locally path connected if the group is not finitely generated. The theorem is proved by an explicit construction of an appropriate space X _G for every countable group G.     

Categorically Proper Homomorphisms of Topological Groups

We extend the Dikranjan-Uspenskij notions of c-compact and h-complete topological group to the morphism level, study the stability properties of the newly defined types of maps, such as closure under direct products, and compare them with their counterparts in topology. We assume Hausdorffness only when our proofs require us to do so, which leads to new results and the affirmation of some facts that were known in a Hausdorff context.     

广义同态与算子群

设G1,G2是群,映射f:G1→G2叫作G1到G2的广义同态映射,如果a,b∈G1,等式(ab)f=afbf和(ab)f=bfaf至少有一个成立.利用广义同态映射的概念,本文将算子群的算子集进行扩充,得到一系列有关算子群的结果,从而推广经典的算子群理论.     

Homomorphisms of Edge-Colored Graphs and Coxeter Groups

Let be two edge-colored graphs (without multiple edges or loops). A homomorphism is a mapping : for which, for every pair of adjacent vertices u and v of G ₁, (u) and (v) are adjacent in G ₂ and the color of the edge (u)(v) is the same as that of the edge uv.We prove a number of results asserting the existence of a graphG , edge-colored from a set C, into which every member from a given class of graphs, also edge-colored from C, maps homomorphically.We apply one of these results to prove that every three-dimensional hyperbolic reflection group, having rotations of orders from the setM ={m₁, m₂,..., m_k}, has a torsion-free subgroup of index not exceeding some bound, which depends only on the setM .     

Expansive Algebraic Actions of Countable Abelian Groups

This paper gives an algebraic characterization of expansive actions of countable abelian groups on compact abelian groups. This naturally extends the classification of expansive algebraic -actions given by Schmidt using complex varieties. Also included is an application to a natural class of examples arising from unit subgroups of integral domains. The author is very grateful for the helpful advice and comments of Tom Ward.     

On Fully Idempotent Homomorphisms of Abelian Groups

Siberian Mathematical Journal - We provide some examples of irregular fully idempotent homomorphisms and study the pairs of abelian groups A and B for which the homomorphism group Hom(A, B) is...     

Determination of Homomorphisms between Linear Groups of the same Degree over Division Rings

We determine all group homomorphisms SL_n(D) GL_n(E), where Dand E are division rings such that char D = char E and E isfinite-dimensional over its centre. Apart from the case wheren = 2 and D is F₂ or F_3, all non-trivial group homomorphismsSL_n(D) GL_n(E) are induced from homomorphisms and anti-homomorphismsD E.     

On Homomorphisms from Semigroups onto Cyclic Groups

   Communicated by     

Definability of Completely Decomposable Torsion-Free Abelian Groups by Groups of Homomorphisms

Let C be an Abelian group. An Abelian group A in some class of Abelian groups is said to be _C H-definable in the class if, for any group B\in , it follows from the existence of an isomorphism Hom(C,A) Hom(C,B) that there is an isomorphism A B. If every group in is _C H-definable in , then the class is called an _C H-class. In the paper, conditions are studied under which a class of completely decomposable torsion-free Abelian groups is a _C H-class, where C is a completely decomposable torsion-free Abelian group.     

可数sofic群的等距线性作用的维数

我们对复Banach空间上的可数sofic群的等距线性作用提出了一种新的维数,推广了复Banach空间上的可数顺从群的等距线性作用的Voiculescu维数,并且在可数sofic群的情形回答了Gromov的一个问题.     

Homomorphisms between Specht modules

In positive characteristic, the Specht modules S corresponding to partitions are not necessarily irreducible, and understanding their structure is vital to understanding the representation theory of the symmetric group. In this paper, we address the related problem of finding the spaces of homomorphisms between Specht modules. Indeed in [2], Carter and Payne showed that the space of homomorphisms from S to S is non-zero for certain pairs of partitions and where the Young diagram for is obtained from that for by moving several nodes from one row to another. We also consider partitions of this type, and, by explicitly examining certain combinations of semi-standard homomorphisms, we are able to give a constructive proof of the Carter–Payne theorem and to generalise it.Mathematics Subject Classification (2000): 20C30     

Homomorphisms of graphs into odd cycles

We give a class of graphs G for which there exists a homomorphism (= adjacency preserving map) from V(G) to V(C), where C is the shortest odd cycle in G, thereby extending a result of Albertson, Catlin, and Gibbons. Our class of graphs is characterized by the following property: For each odd subdivision G′ of G there exists a homomorphic map from V(G′) to V(C), where C′ is the shortest odd cycle of G′.