Compact open sets in dual spaces and projections in group algebras of [FC]? groups

The structure of a compact open set in the dual of an [FC]^? group G, a locally compact group with relatively compact conjugacy classes, is given in terms of certain subsets which arise somewhat naturally. The support in the dual of a projection in L ¹(G) is a compact open set. Therefore, knowledge of the structure of such sets helps in identifying and constructing projections. We describe explicitly the compact open sets and construct projections for some illustrative examples.     

Sylow Theory for Quasigroups II: Sectional Action

The first paper in this series initiated a study of Sylow theory for quasigroups and Latin squares based on orbits of the left multiplication group. The current paper is based on so‐called pseudo‐orbits, which are formed by the images of a subset under the set of left translations. The two approaches agree for groups, but differ in the general case. Subsets are described as sectional if the pseudo‐orbit that they generate actually partitions the quasigroup. Sectional subsets are especially well behaved in the newly identified class of conflatable quasigroups, which provides a unified treatment of Moufang, Bol, and conjugacy closure properties. Relationships between sectional and Lagrangean properties of subquasigroups are established. Structural implications of sectional properties in loops are investigated, and divisors of the order of a finite quasigroup are classified according to the behavior of sectional subsets and pseudo‐orbits. An upper bound is given on the size of a pseudo‐orbit. Various interactions of the Sylow theory with design theory are discussed. In particular, it is shown how Sylow theory yields readily computable isomorphism invariants with the resolving power to distinguish each of the 80 Steiner triple systems of order 15.     

Orbit Equivalence and Topological Conjugacy of Affine Actionson Compact Abelian Groups

 We study topological rigidity of affine actions on compact connected metrizable abelian groups. We also classify one parameter flows of translations up to orbit equivalence and discrete group actions by translations up to topological conjugacy. (Received 21 December 1998; in revised form 2 June 1999)     

A generalized notion of quotient for finite Abelian groups

A finite Abelian group G is partitioned into subsets which are translations of each other. A binary operation is defined on these sets in a way which generalizes the quotient group operation. Every finite Abelian group can be realized as such a generalized quotient with G cyclic.     

Splitting fields of characteristic polynomials of random elements in arithmetic groups

We discuss rather systematically the principle, implicit in earlier works, that for a “random” element in an arithmetic subgroup of a (split, say) reductive algebraic group over a number field, the splitting field of the characteristic polynomial, computed using any faitfhful representation, has Galois group isomorphic to the Weyl group of the underlying algebraic group. Besides tools such as the large sieve, which we had already used, we introduce some probabilistic ideas (large deviation estimates for finite Markov chains) and the general case involves a more precise understanding of the way Frobenius conjugacy classes are computed for such splitting fields (which is related to a map between regular elements of a finite group of Lie type and conjugacy classes in the Weyl group which had been considered earlier by Carter and Fulman for other purposes; we show in particular that the values of this map are equidistributed).     

On groups of odd order with exactly two non-central conjugacy classes of each size

The non-abelian group of order 21 is the only finite group of odd order with exactly two non-central conjugacy classes of each size. Received: 31 January 2005     

Second-order subelliptic operators on Lie groups III: Hölder continuous coefficients

Let G be a connected Lie group with Lie algebra and an algebraic basis of . Further let denote the generators of left translations, acting on the -spaces formed with left Haar measure dg, in the directions . We consider second-order operators corresponding to a quadratic form with complex coefficients , , , . The principal coefficients are assumed to be H?lder continuous and the matrix is assumed to satisfy the (sub)ellipticity condition uniformly over G. We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel. Moreover, we establish Gaussian type bounds for the kernel and its derivatives. Similar theorems are proved for operators in nondivergence form for which the principal coefficients are at least once differentiable. Received January 24, 1997 / Accepted June 5, 1998     

Projective ring line encompassing two-qubits

We find that the projective line over the (noncommutative) ring of 2×2 matrices with coefficients in GF(2) fully accommodates the algebra of 15 operators (generalized Pauli matrices) characterizing two-qubit systems. The relevant subconfiguration consists of 15 points, each of which is either simultaneously distant or simultaneously neighbor to (any) two given distant points of the line. The operators can be identified one-to-one with the points such that their commutation relations are exactly reproduced by the underlying geometry of the points with the ring geometric notions of neighbor and distant corresponding to the respective operational notions of commuting and noncommuting. This remarkable configuration can be viewed in two principally different ways accounting for the basic corresponding 9+6 and 10+5 factorizations of the algebra of observables: first, as a disjoint union of the projective line over GF(2) × GF(2) (the “Mermin” part) and two lines over GF(4) passing through the two selected points that are omitted; second, as the generalized quadrangle of order two with its ovoids and/or spreads corresponding to (maximum) sets of five mutually noncommuting operators and/or groups of five maximally commuting subsets of three operators each. These findings open unexpected possibilities for an algebro-geometric modeling of finite-dimensional quantum systems and completely new prospects for their numerous applications. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 155, No. 3, pp. 463–473, June, 2008.     

First-order definability and algebraicity of the sets of annihilating and generating collections of elements for some relatively free solvable groups

We study the first-order definable, Diophantine, and algebraic subsets in the set of all ordered sets generating a group or generating a group as a normal subgroup for some relatively free solvable groups.     

On left conjugacy closed loops in which the left multiplication group is normal

A loopQ is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. This papers investigates those LCC loops where the group generated by left translations is normal in the group generated by both left and right translations.     

Infinite co-minimal pairs involving lacunary sequences and generalisations to higher dimensions

In 2011, Nathanson proposed several questions on minimal complements in a group or a semigroup. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the authors. In this article, we study which type of subsets in the integers and free abelian groups of higher rank can be a part of a co-minimal pair. We show that a majority of lacunary sequences have this property. From the conditions established, one can show that any infinite subset of any finitely generated abelian group has uncountably many subsets which is a part of a co-minimal pair. Further, the uncountable collection of sets can be chosen so that they satisfy certain algebraic properties.

     

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.     

FC groups whose periodic parts can be embedded in direct products of finite groups

In this note are considered FC groups whose periodic parts can be embedded in direct products of finite groups. It is shown that if the periodic part of an FC group G can be embedded in the direct product of its finite factor groups with respect to the normal subgroups of G whose intersection is the trivial subgroup, then G/Z (G) is a subgroup of a direct product of finite groups. It is also shown that if the periodic part of an FC group G is a group without a center, then G can be embedded in a direct product of finite groups without centers and a torsion-free Abelian group. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 9–20, January, 1977. The author is thankful to the referee for making many valuable remarks.     

On the geometry of conjugacy classes in classical groups

Summary We study closures of conjugacy classes in the Lie algebras of the orthogonal and symplectic groups and determine which ones are normal varieties. Furthermore we give a complete classification of the minimal singularities which arise in this context, i.e. the singularities which occur in the open classes in the boundary of a given conjugacy class. In contrast to the results for the general linear group ([KP1], [KP2]) there are classes with non normal closure; they are branched in a class of codimension two and give rise to normal minimal singularities. The methods used are (classical) invariant theory and algebraic geometry. Supported in part by the SFB Theoretische Mathematik, University of Bonn, and by the University of Hamburg     

On left conjugacy closed loops with a nucleus of index two

A loop Q is said to be left conjugacy closed (LCC) if the left translations form a set of permutations that is closed under conjugation. Loops in which the left and middle nuclei coincide and are of index 2 are necesarilly LCC, and they are constructed in the paper explicitly. LCC loops Q with the right nucleus G of index 2 offer a larger diversity, but that is associated with the level of commutativity of G (amongst others, the centre of G has to be nontrivial). On the other hand, for each m ≥ 2 one can construct an LCC loop Q of order 2m in such a way that its left nucleus is trivial, and the right nucleus if of order m. If Q is involutorial, then it is a Bol loop. Work supported by institutional grant MSM 113200007 and by Grant Agency of Czech Republic, Grant 201/02/0594. The paper was written while the author was visiting Universitaet Hamburg in January 2004.     

The monoid of queue actions

We model the behavior of a fifo-queue as a monoid of transformations that are induced by sequences of writing and reading. We describe this monoid by means of a confluent and terminating semi-Thue system and study some of its basic algebraic properties such as conjugacy. Moreover, we show that while several properties concerning its rational subsets are undecidable, their uniform membership problem is \({{\mathsf {N}}}{{\mathsf {L}}}\)-complete. Furthermore, we present an algebraic characterization of this monoid’s recognizable subsets. Finally, we prove that it is not Thurston-automatic.     

On a theorem by Benson and Conway

We define some purely lattice theoretic translations of algebraic notions related to submodule lattices, leading to new structural features of modular lattices and to generalisations of the Benson-Conway Theorem.     

On the Loewy Length and the Noetherian Dimension of Artinian Modules

The problem of computing the automorphism groups of an elementary Abelian Hadamard difference set or equivalently of a bent function seems to have attracted not much interest so far. We describe some series of such sets and compute their automorphism group. For some of these sets the construction is based on the nonvanishing of the degree 1-cohomology of certain Chevalley groups in characteristic two. We also classify bent functions f such that Aut(f) together with the translations from the underlying vector space induce a rank 3 group of automorphisms of the associated symmetric design. Finally, we discuss computational aspects associated with such questions.