We exhibit a variation of the Lazard Elimination Theorem for free restricted Lie algebras, and apply it to two problems about finite group actions on free Lie algebras over fields of positive characteristic.
Using a result on arithmetic progressions, we describe a method for finding the rational h-tuples rho = (rho (1),..., p(h),) such that all the multiples m rho (for m coprime to a denominator of rho) lie in a linear variety module Z. We give an application to hypergeometric functions.
Abstract By blending techniques from set theory and algebraic topology we investigate the order of any homeomorphism of the nth power of the long ray or long line L having finite order, finding all possible orders when n = 1, 2, 3 or 4 in the first case and when n = 1 or 2 in the second. We also show that all finite powers of L are acyclic with respect to Alexander-Spanier cohomology.
AbstractA class of totally disconnected groups consisting of partial direct products on an index set is examined. For such a group, the scale function is found, and for automorphisms arising from permutations of the index set, the tidy subgroups are characterised. When applied to the case where the index set is a finitely-generated free group and the permutation is translation by an element x of the group, the scale depends on the cyclically reduced form of x and the tidy subgroup on the element which conjugates x to its cyclically reduced form.
Abstract A pseudo-Riemannian manifold is said to be timelike (spacelike) Osserman if the Jordan form of the Jacobi operator Kx is independent of the particular unit timelike (spacelike) tangent vector X. The first main result is that timelike (spacelike) Osserman manifold (M, g) of signature (2, 2) with the diagonalizable Jacobi operator is either locally rank-one symmetric or flat. In the nondiagonalizable case the characteristic polynomial of Kx has to have a triple zero, which is the other main result. An important step in the proof is based on Walker's study of pseudo-Riemannian manifolds admitting parallel totally isotropic distributions. Also some interesting additional geometric properties of Osserman type manifolds are established. For the nondiagonalizable Jacobi operators some of the examples show a nature of the Osserman condition for Riemannian manifolds different from that of pseudo-Riemannian manifolds.
We find a basis for the universal punctured even distribution and then a basis for the cyclotomic units over function fields.
Abstract This paper inverstigates the automorphism groups of Cayley graphs of metracyclic p-gorups. A characterization is given of the automorphism groups of Cayley grahs of a metacyclic p-group for odd prime p. In particular, a complete determiniation of the automophism group of a connected Cayley graph with valency less than 2p of a nonabelian metacyclic p-group is obtained as a consequence. In subsequent work, the result of this paper has been applied to solve several problems in graph theory.
We show that if G is a finitely generated profinite group such that [x(1),x(2),...x(k)] is Engel for any x(1),x(2),...,x(k) is an element of G then gamma (k)(G) is locally nilpotent, and if [x(1),x(2),...,x(k)] has finite order for any x(1),x(2),...,x(k) is an element of G then, under some additional assumptions, gamma (k)(G) is locally finite.
AbstractIn this note we investigate lacunarity or ‘thin’ subsets in the dual object of a compact group via different classes of summing operators between Banach spaces. In particular, we give characterisations of Sidon and ∧ (p) sets, 2 < p < ∞
Bounds are obtained for the minimum number of generators for the fundamental groups of a fan-Lily of closed 3-dimensional manifolds. A significant role has been played by the use of computers.
An asymptotic estimate is derived for the expected number of extrema of a polynomial a(0) + a(1)((n)(1))(1/2)x + a(2)((n)(2))(1/2)x(2) + ... + a(n)((n)(n))(1/2)x(n) whose independent normal coefficients possess non-equal non-zero mean values. A result is presented that generalizes in terms of normal processes the analytical device used for construction of similar asymptotic estimates for random polynomials with normal coefficients.
There are at least three imprimitivity bimodules naturally associated to a maximal coaction of a discrete group G on a C*-algebra and a normal subgroup of G: Mansfield's bimodule; the bimodule assembled by Ng from Green's imprimitivity bimodule and Katayama duality; and a bimodule assembled from Green's bimodule and a crossed-product Mansfield bimodule. We show that all three of these are isomorphic, so that the corresponding inducing maps on representations are identical. This can be interpreted as saying that Mansfield and Green induction are inverses of one another ``modulo Katayama duality''. These results pass to twisted coactions; dual results starting with an action are also given.
Let S be a subset of a group G such that S-1 = S. Denote by gr(S) the subgroup of G generated by S, and by ls(g) the length of an element g is an element of gr(S) relative to the set S. Suppose that V is a finite subset of a free group F of countable rank such that the verbal subgroup V(F) is a proper subgroup of F. For an arbitrary group G, denote by (V) over bar (G) the set of values in G of all the words from the set V. In the present paper, for amalgamated products G = A *(H) B such that A not equal H and the number of double cosets of B by H is at least three, the infiniteness of the set {l(S)(g) \ g is an element of gr(S)}, where S = (V) over bar (G) boolean OR (V) over bar (G)(-1), is established.
Abstract A subset F of an ordered set X is a fibre of X if F intersects every maximal antichain of X. We find a lower bound on the function ƒ (D), the minimum fibre size in the distributive lattice D, in terms of the size of D. In particular, we prove that there is a constant c such that In the process we show that minimum fibre size is a monotone property for a certain class of distributive lattices. This fact depends upon being able to split every maximal antichain of this class of distributive lattices into two parts so that the lattice is the union of the upset of one part and the downset of the other.
AbstractIn this paper we give a complete description of diameter-preserving linear bijections on the space of affine continuous functions on a compact convex set whose extreme points are split faces. We also give a description of such maps on function algebras considered on their maximal ideal space. We formulate and prove similar results for spaces of vector-valued functions.
AbstractThe main result is that every torsion-free locally nilpotent group that is isomorphic to each of its nonnilpotent subgroups is nilpotent, that is, a torsion-free locally nilpotent group G that is not nilpotent has a non-nilpotent subgroup H that is not isomorphic to G.
Let G be a finite group of order p(k), where p is a prime and k greater than or equal to 1, such that G is either cyclic, quaternion or generalised quaternion. Let V be a finite-dimensional free KG-module where K is a field of characteristic p. The Lie powers L-n( V) are naturally KG-modules and the main result identifies. these modules up to isomorphism. There are only two isomorphism types of indecomposables occurring as direct summands of these modules, namely the regular KG-module and the indecomposable of dimension p(k) - p(k-1) induced from the indecomposable KH-module of dimension p - 1, where H is the unique subgroup of G of order p. Formulae are given for the multiplicities of these indecomposables. in L-n( V). This extends and utilises work of the first author and R. Stohr concerned with the case where G has order p.
Let G be a finite group that acts on a finite group V, and let p be a prime that does not divide the order of V. Then the p-parts of the orbit sizes are the same in the actions of G on the sets of conjugacy classes and irreducible characters of V. This result is derived as a consequence of some general theory relating orbits and chains of p-subgroups of a group.
AbstractGiven polynomialsaandbover an integral domainR, their tensor product (denoteda ⊗ b) is a polynomial overRof degree deg(a) deg(b) whose roots comprise all products αβ, where α is a root of a, and β is a root ofb. This paper considers basic properties of ⊗ including how to factora ⊗ binto irreducibles factors, and the direct sum decomposition of the ⊗-product of fields.
We investigate certain norm and continuity conditions that provide us with 'unique Hahn-Banach Theorems' from P((n)c(0)) to P((n)l(infinity)) and from P-N(E-n) to P(E-n "). We show that there is a unique norm-preserving extension for norm-attaining 2-homogeneous polynomials on complex co to a, but there is no unique norm-presenting extension from P((3)c(0)) to P((3)l(infinity)).
