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.     

Each Invertible Sharply d-Transitive Finite Permutation Set with d ≥ 4 is a Group

All known finite sharply 4-transitive permutation sets containing the identity are groups, namely S ₄, S ₅, A ₆ and the Mathieu group of degree 11. We prove that a sharply 4-transitive permutation set on 11 elements containing the identity must necessarily be the Mathieu group of degree 11. The proof uses direct counting arguments. It is based on a combinatorial property of the involutions in the Mathieu group of degree 11 (which is established here) and on the uniqueness of the Minkowski planes of order 9 (which had been established before): the validity of both facts relies on computer calculations. A permutation set is said to be invertible if it contains the identity and if whenever it contains a permutation it also contains its inverse. In the geometric structure arising from an invertible permutation set at least one block-symmetry is an automorphism. The above result has the following consequences. i) A sharply 5-transitive permutation set on 12 elements containing the identity is necessarily the Mathieu group of degree 12. ii) There exists no sharply 6-transitive permutation set on 13 elements. For d 6 there exists no invertible sharply d-transitive permutation set on a finite set with at least d + 3 elements. iii) A finite invertible sharply d-transitive permutation set with d 4 is necessarily a group, that is either a symmetric group, an alternating group, the Mathieu group of degree 11 or the Mathieu group of degree 12.     

Kazhdan's Property (T) for the unitary group of aseparable Hilbert space

We show that the unitary group of a separable Hilbert space has Kazhdan's Property (T), when it is equipped with the strong operator topology. More precisely, for every integer m 2, we give an explicit Kazhdan set consisting of m unitary operators and determine an optimal Kazhdan constant for this set. Moreover, we show that a locally compact group with Kazhdan's Property (T) has a finite Kazhdan set if and only if its Bohr compactification has a finite Kazhdan set. As a consequence, if a locally compact group with Property (T) is minimally almost periodic, then it has a finite Kazhdan set.     

Hamiltonian cycles in cayley color graphs

A group Γ is said to possess a hamiltonian generating set if there exists a minimal generating set Δ for Γ such that the Cayley color graph D_Δ(Γ) is hamiltonian. It is shown that every finite abelian group has a hamiltonian generating set. Certain classes of nonabelian groups are also investigated.     

Diffeomorphism classification of finite group actions on disks

In this paper we discuss the diffeomorphism classification of finite group actions on disks. We answer the question when an action on a space M can be extended to an action on a disk such that the action is free away from M. Let the singular set consist of the points with nontrivial isotropy group. We show (under some dimension assumptions) that disks with diffeomorphic neighborhoods of the singular set can be imbedded into each other. As a consequence we find a classification of group actions on disks in terms of the neighborhood of the singular set and an element in the Whitehead group of G.     

An Infinite Presentation of the Torelli Group

In this paper, we construct an infinite presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all “separating twists”, all “bounding pair maps”, and all “commutators of simply intersecting pairs” and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of all simple closed curves.     

Groups definable in local fields and pseudo-finite fields

Using model-theoretic methods we prove: Theorem A If G is a Nash group over the real or p-adic field, then there is a Nash isomorphism between neighbourhoods of the identity of G and of the set of F-rational points of an algebraic group defined over F. Theorem B Let G be a connected affine Nash group over ℝ. Then G is Nash isogeneous with the (real) connected component of the set of real points of an algebraic group defined over ℝ. Theorem C Let G be a group definable in a pseudo-finite field F. Then G is definably virtually isogeneous with the set of F-rational points of an algebraic group defined over F. Both authors supported by NSF grants.     

Functors between shape categories

Permutation groups of prime power degree are investigated here through the study of the corresponding group algebra of the set of all functions from the underlying set on which the permutation group acts to a finite field of characteristic p. For the case when the permutation group is of degree p² acting on a set consisting of the direct product of two elementary abelian p-groups, the structure of a minimal permutation module is obtained under certain conditions. The proofs do not depend on the recent classification results of finite simple groups.     

The direction of an automorphism of a totally disconnected locally compact group

We describe the asymptotic behavior of automorphisms of totally disconnected locally compact groups in terms of a set of `directions' which comes equipped with a natural pseudo-metric. The structure at infinity obtained by completing the induced metric quotient space of the set of directions recovers familiar objects such as: the set of ends of the tree for the group of inner automorphisms of the group of isometries of a regular locally finite tree; and the spherical Bruhat-Tits building for the group of inner automorphisms of the set of rational points of a semisimple group over a local field. Research supported by A.R.C. Grant DP0208137.     

组合的分组测试算法的近似控制标准

对于给定的一个集合,分组测试问题是通过一系列的测试去确定这个集合的一个子集. 在文中, 作者首先运用动态规划的理论与方法, 建立了一个近似控制标准, 目的是对分组测试算法的构建过程进行有效控制, 使所构建的算法达到最优. 其次, 应用该近似控制标准研究了在n个硬币集合中确定一个伪硬币的最小平均测试数的问题. 文中所涉及的近似控制问题, 给出了在一个给定集合中去确定这个集合的一个子集的最优分组测试算法, 该最优分组测试算法是在平均测试步骤最少意义下的最优分组测试算法.     

On dynamics of Out(F n ) on PSL2(ℂ) characters

We introduce and study an open set of PSL₂(?) characters of a nonabelian free group, on which the action of the outer automorphism group is properly discontinuous, and which is strictly larger than the set of discrete, faithful convex-cocompact (i.e. Schottky) characters. This implies, in particular, that the outer automorphism group does not act ergodically on the set of characters with dense image. Hence there is a difference between the geometric (discrete vs. dense) decomposition of the characters, and a natural dynamical decomposition.     

The rational group algebra of a normally monomial group

In this paper, a complete irredundant set of a class of strong Shoda pairs of a finite group G is computed. The algebraic structure of the rational group algebra of a normally monomial group is thus obtained. A necessary and sufficient condition for G to be normally monomial is derived. The main result is also illustrated by computing a complete set of primitive central idempotents and the explicit Wedderburn decomposition of the rational group algebra of some normally monomial groups.     

Abelian subgroups generated by Dehn twists in homeomorphism group

The subgroup of homeoniorphisni group generated by Dehn twists along the set of simple closed pairwise non-homotopic curves with some conditions is studied. It is proved that this group is isomorphic to a free Abelian group of rank k, where k is the number of curves in the set. In the case of an orientable surface, this result is classical.     

Commutativity of localized self-homotopy groups of symplectic groups

The self-homotopy group of a topological group G is the set of homotopy classes of self-maps of G equipped with the group structure inherited from G. We determine the set of primes p such that the p-localization of the self-homotopy group of Sp(n) is commutative. As a consequence, we see that this group detects the homotopy commutativity of p-localized Sp(n) by its commutativity almost all cases.     

Cocliques of maximal size in the prime graph of a finite simple group

A prime graph of a finite group is defined in the following way: the set of vertices of the graph is the set of prime divisors of the order of the group, and two distinct vertices r and s are joined by an edge if there is an element of order rs in the group. We describe all cocliques of maximal size for finite simple groups.     

Rational sets in finitely generated nilpotent groups

We deal with a class of rational subsets of a group, that is, the least class of its subsets which contains all finite subsets and is closed under taking union. a product of two sets, and under generating of a submonoid by a set. It is proved that the class of rational subsets of a finitely generated nilpotent group G is a Boolean algebra iff G is Abelian-by-finite. We also study the question asking under which conditions the set of solutions for equations in groups will be rational. It is shown that the set of solutions for an arbitrary equation in one variable in a finitely generated nilpotent group of class 2 is rational. And we give an example of an equation in one variable in a free nilpotent group of nilpotency class 3 and rank 2 whose set of solutions is not rational. Supported by RFFR grant No. 98-01-00932. Translated fromAlgebra i Logika, Vol. 39, No. 4, pp. 379–394, July–August, 2000.     

Representations into the hyperspace of a compact group

In [4], Trnková proved that every commutative semigroup is embeddable in the power set of a commutative group under set product. In this paper, we will discuss the topological aspect of it. One such theorem is that the min interval is not embeddable in the hyperspace of a compact group.     

Weak Uniquely Completable Sets for Finite Groups

Keedwell has shown that none of the groups of order less than5 has a weak uniquely completable set. We prove that a weakuniquely completable set exists in a latin square based on afinite group if and only if the group is of order greater than5. 1991 Mathematics Subject Classification 05B15.     

Springer correspondence for disconnected groups

The Springer correspondence is a map from the set of unipotent conjugacy classes of a reductive algebraic group to the set of irreducible complex characters of the Weyl group. Here, we determine this map explicitly in the case of disconnected classical algebraic groups. Mathematics Subject Classification (2000): Primary 20G05; Secondary 20C33.