Nonlinearity of the Automorphism Groups of Some Free Algebras

We prove that the groups of tame automorphisms of a free Lie algebra (free associative algebra, absolutely free algebra, algebra of polynomials) of rank at least four over a field of characteristic zero admit no faithful representation by matrices over any field.     

On the Bergman Property for the Automorphism Groups of Relatively Free Groups

A group G is said to have the Bergman property (the propertyof uniformity of finite width) if given any generating X withX = X^–1 of G, we have that G = X^k for some natural k,that is, every element of G is a product of at most k elementsof X. We prove that the automorphism group Aut(N) of any infinitelygenerated free nilpotent group N has the Bergman property. Also,we obtain a partial answer to a question posed by Bergman byestablishing that the automorphism group of a free group ofcountably infinite rank is a group of uniformly finite width.     

On Homomorphisms from Semigroups onto Cyclic Groups

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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.     

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.     

Characterizing Automorphism Groups of Ordered Abelian Groups

A proof is given of the following theorem, which characterizesfull automorphism groups of ordered abelian groups: a groupH is the automorphism group of some ordered abelian group ifand only if H is right-orderable. 2000 Mathematics Subject Classification20K15, 20K20, 20F60, 20K30 (primary); 03E05 (secondary).     

Recognizing the Automorphism Groups of Mathieu Groups

It is a well-known fact that characters of a finite group can give important information about the structure of the group. It was also proved by the third author that a finite simple group can be uniquely determined by its character table. Here the authors attempt to investigate how to characterize a finite almost-simple group by using less information of its character table, and successfully characterize the automorphism groups of Mathieu groups by their orders and at most two irreducible character degrees of their character tables.     

Homomorphisms between Stone-Cech Homomorphisms into Stone-Cech Remainders of Countable Groups

We study properties of continuous homomorphisms from β S into T^* and from S^* into T^*, where S denotes a countably infinite semigroup and T denotes a countably infinite group. We show that they have striking algebraic properties if they do not arise as continuous extensions of homomorphisms from S to T.     

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.     

All Groups are Outer Automorphism Groups of Simple Groups

It is shown that each group is the outer automorphism groupof a simple group. Surprisingly, the proof is mainly based onthe theory of ordered or relational structures and their symmetrygroups. By a recent result of Droste and Shelah, any group isthe outer automorphism group Out (Aut T) of the automorphismgroup Aut T of a doubly homogeneous chain (T, ). However, AutT is never simple. Following recent investigations on automorphismgroups of circles, it is possible to turn (T, ) into a circleC such that Out (Aut T) Out (Aut C). The unavoidable normalsubgroups in Aut T evaporate in Aut C, which is now simple,and the result follows.     

Automorphism Groups of Schur Rings

In 1993, Muzychuk [23 Muzychuk , Mikhail E. ( 1993 ). The structure of rational Schur rings over cyclic groups . European Journal of Combinatorics 14 : 479 – 490 .[Crossref], [Web of Science ®] , [Google Scholar]] showed that the rational Schur rings over a cyclic group Z _n are in one-to-one correspondence with sublattices of the divisor lattice of n, or equivalently, with sublattices of the lattice of subgroups of Z _n . This can easily be extended to show that for any finite group G, sublattices of the lattice of characteristic subgroups of G give rise to rational Schur rings over G in a natural way. Our main result is that any finite group may be represented as the (algebraic) automorphism group of such a rational Schur ring over an abelian p-group, for any odd prime p. In contrast, over a cyclic group the automorphism group of any Schur ring is abelian. We also prove a converse to the well-known result of Muzychuk [24 Muzychuk , Mikhail E. ( 1994 ). On the structure of basic sets of Schur rings over cyclic groups . Journal of Algebra 169 : 655 – 678 .[Crossref], [Web of Science ®] , [Google Scholar]] that two Schur rings over a cyclic group are isomorphic if and only if they coincide; namely, we show that over a group which is not cyclic, there always exist distinct isomorphic Schur rings.     

Automorphism Groups of Moishezon Threefolds

Mathematical Notes -     

On Orbits of Automorphism Groups

If G is a finite group and if A is a group of automorphisms of G whose fixed point subgroup is C _G (A) then every subgroup F of C _G (A) acts on the set of orbits of A in G. The peculiarities of this action are used here to derive several results on the number of orbits of A in an economical manner.Original Russian Text Copyright © 2005 Deaconescu M. and Walls G. L.__________Translated from Sibirskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 533–537, May–June, 2005.     

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.     

Structure of Irreducible Homomorphisms to/from Free Modules

The primary goal of this paper is to investigate the structure of irreducible monomorphisms to and irreducible epimorphisms from finitely generated free modules over a noetherian local ring. Then we show that over such a ring, self-vanishing of Ext and Tor for a finitely generated module admitting such an irreducible homomorphism forces the ring to be regular.     

广义四元数群的Coleman外自同构群

在这篇注记中,我们利用群的射影极限性质证明了广义四元数群的Coleman外自同构群或者是1或者是一个初等阿贝尔2-群.     

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.