Some corollaries of Frobenius' normal -complement theorem

For a prime divisor of the order of a finite group , we present the set of -subgroups generating . In particular, we present the set of primary subgroups of generating the last member of the lower central series of . The proof is based on the Frobenius Normal -Complement Theorem and basic properties of minimal nonnilpotent groups. Let be a group and a group-theoretic property inherited by subgroups and epimorphic images such that all minimal non--subgroups (-subgroups) of are not nilpotent. Then (see the lemma), if is generated by all -subgroups of it follows that is a -group.

     

-identities on associative algebras

Let be an algebra over a field and a finite group of automorphisms and anti-automorphisms of . We prove that if satisfies an essential -polynomial identity of degree , then the -codimensions of are exponentially bounded and satisfies a polynomial identity whose degree is bounded by an explicit function of . As a consequence we show that if is an algebra with involution satisfying a -polynomial identity of degree , then the -codimensions of are exponentially bounded; this gives a new proof of a theorem of Amitsur stating that in this case must satisfy a polynomial identity and we can now give an upper bound on the degree of this identity.

     

On zeros of characters of finite groups

We present several results connecting the number of conjugacy classes of a finite group on which an irreducible character vanishes, and the size of some centralizer of an element. For example, we show that if is a finite group such that , then has an element , such that , where is the maximal number of zeros in a row of the character table of . Dual results connecting the number of irreducible characters which are zero on a fixed conjugacy class, and the degree of some irreducible character, are included too. For example, the dual of the above result is the following: Let be a finite group such that ; then has an irreducible character such that , where is the maximal number of zeros in a column of the character table of .

     

On permutation representations of polyhedral groups

We answer affirmatively the following question of Derek Holt: Given integers , can one, in a simple manner, find a finite set and permutations such that has order , has order and has order ? The method of proof enables us to prove more general results (Theorems 2 and 3).

     

On sums and products of integers

Erdös and Szemerédi proved that if is a set of positive integers, then there must be at least integers that can be written as the sum or product of two elements of , where is a constant and . Nathanson proved that the result holds for . In this paper it is proved that the result holds for and .

     

Group rings whose symmetric elements are Lie nilpotent

Let be the group ring of a group over a field , with characteristic different from . Let denote the natural involution on sending each group element to its inverse. Denote by the set of symmetric elements with respect to this involution. A paper of Giambruno and Sehgal showed that provided has no -elements, if is Lie nilpotent, then so is . In this paper, we determine when is Lie nilpotent, if does contain -elements.

     

Schreier theorem on groups which split over free abelian groups

Let be either a free product with amalgamation or an HNN group where is isomorphic to a free abelian group of finite rank. Suppose that both and have no nontrivial, finitely generated, normal subgroups of infinite indices. We show that if contains a finitely generated normal subgroup which is neither contained in nor free, then the index of in is finite. Further, as an application of this result, we show that the fundamental group of a torus sum of -manifolds and , the interiors of which admit hyperbolic structures, have no nontrivial, finitely generated, nonfree, normal subgroup of infinite index if each of and has at least one nontorus boundary.

     

Higher order symmetric spaces and the roots of the identity in a Lie group

Let denote the set of all -roots of the identity in a Lie group . We show that is always an embedded submanifold of , having the conjugacy classes of its elements as open submanifolds. These conjugacy classes are examples of -symmetric spaces and we show, more generally, that every -symmetric space of a Lie group is a covering manifold of an embedded submanifold of . We compute also the Hessian of the inclusions of and into , relative to the natural connection on the domain and to the symmetric connection on .

     

Normalizers of the congruence subgroups of the Hecke group

Let cos and let be the Hecke group associated to . In this article, we show that for a prime ideal in , the congruence subgroups of are self-normalized in .

     

Sharper changes in topologies

Let be a Polish group, a Polish topology on a space , acting continuously on , with -invariant and in the Borel algebra generated by . Then there is a larger Polish topology on so that is open with respect to , still acts continuously on , and has a basis consisting of sets that are of the same Borel rank as relative to .

     

A trace formula for Hankel operators

We show that if is an operator valued analytic function in the open right half plane such that the Hankel operator with symbol is of trace-class, then has continuous extension to the imaginary axis,

exists in the trace-class norm, and .

     

Reduction in principal fiber bundles: Covariant Euler-Poincaré equations

Let be a principal -bundle, and let be a -invariant Lagrangian density. We obtain the Euler-Poincaré equations for the reduced Lagrangian defined on , the bundle of connections on .

     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

Helly-type theorems for hollow axis-aligned boxes

A hollow axis-aligned box is the boundary of the cartesian product of compact intervals in . We show that for , if any of a collection of hollow axis-aligned boxes have non-empty intersection, then the whole collection has non-empty intersection; and if any of a collection of hollow axis-aligned rectangles in have non-empty intersection, then the whole collection has non-empty intersection. The values for and for are the best possible in general. We also characterize the collections of hollow boxes which would be counterexamples if were lowered to , and to , respectively.

     

Group algebras with units satisfying a group identity

Let be the group algebra of a group over a field , and let be its group of units. A conjecture by Brian Hartley asserts that if is a torsion group and satisfies a group identity, then satisfies a polynomial identity. This was verified earlier in case is an infinite field. Here we modify the original proof so that it handles fields of all sizes.

     

Smith equivalence of representations for finite perfect groups

Using smooth one-fixed-point actions on spheres and a result due to Bob Oliver on the tangent representations at fixed points for smooth group actions on disks, we obtain a similar result for perfect group actions on spheres. For a finite group , we compute a certain subgroup of the representation ring . This allows us to prove that a finite perfect group has a smooth -proper action on a sphere with isolated fixed points at which the tangent representations of are mutually nonisomorphic if and only if contains two or more real conjugacy classes of elements not of prime power order. Moreover, by reducing group theoretical computations to number theory, for an integer and primes , we prove similar results for the group , , or . In particular, has Smith equivalent representations that are not isomorphic if and only if , , .

     

On the range and the kernel of the operator

Let denote the algebra of (bounded linear) operators on the separable complex Hilbert space , and let denote a norm ideal in . For , let the derivation be defined by , and let be defined by . The main result of this paper is to show that if , are contractions, then for every operator such that , then for all .

     

Centralizers in residually finite torsion groups

Let be a residually finite torsion group. We show that, if has a finite 2-subgroup whose centralizer is finite, then is locally finite. We also show that, if has no -torsion, and is a finite 2-group acting on in such a way that the centralizer is soluble, or of finite exponent, then is locally finite.

     

Toeplitz -algebras on ordered groups and their ideals of finite elements

Let be a discrete abelian group and an ordered group. Denote by the minimal quasily ordered group containing . In this paper, we show that the ideal of finite elements is exactly the kernel of the natural morphism between these two Toeplitz -algebras. When is countable, we show that if the direct sum of -groups , then .

     

On a theorem of Scott and Swarup

Let be an exact sequence of hyperbolic groups induced by an automorphism of the free group . Let be a finitely generated distorted subgroup of . Then there exist and a free factor of such that the conjugacy class of is preserved by and contains a finite index subgroup of a conjugate of . This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.