Quadratic functions and -groups

Properties of -groups are reformulated in terms of quadratic functions and pre-semifields. As a consequence, counter-examples to some earlier results are obtained.

     

Cohomology of buildings and finiteness properties of -groups

Borel and Serre calculated the cohomology of the building associated to a reductive group and used the result to deduce that torsion-free -arithmetic groups are duality groups. By replacing their group-theoretic arguments with proofs relying only upon the geometry of buildings, we show that Borel and Serre's approach can be modified to calculate the cohomology of any locally finite affine building. As an application we show that any finitely presented -group is a virtual duality group. A number of other finiteness conditions for -groups are also established.

     

The extraspecial case of the problem

Let be an extraspecial-type group and a faithful, absolutely irreducible -module, where is a finite field. Let be the normalizer in of . We show that, with few exceptions, there exists a such that the restriction of to is self-dual whenever and .

     

On locally finite -groups satisfying an Engel condition

For a given positive integer and a given prime number , let be the integer satisfying . We show that every locally finite -group, satisfying the -Engel identity, is (nilpotent of -bounded class)-by-(finite exponent) where the best upper bound for the exponent is either or if is odd. When the best upper bound is or . In the second part of the paper we focus our attention on -Engel groups. With the aid of the results of the first part we show that every -Engel -group is soluble and the derived length is bounded by some constant.

     

Fully transitive -groups with finite first Ulm subgroup

An abelian -group is called (fully) transitive if for all with ( ) there exists an automorphism (endomorphism) of which maps onto . It is a long-standing problem of A. L. S. Corner whether there exist non-transitive but fully transitive -groups with finite first Ulm subgroup. In this paper we restrict ourselves to -groups of type , this is to say -groups satisfying . We show that the answer to Corner's question is no if is finite and is of type .

     

Primitive characters of subgroups of -groups

Let be an -group, let be a subnormal subgroup of , and let be a Hall subgroup of . If the character is primitive, then is a power of 2. Furthermore, if is odd, then .

     

Free actions of -groups on products of lens spaces

Let be an odd prime number. We prove that if acts freely on a product of equidimensional lens spaces, then . This settles a special case of a conjecture due to C. Allday. We also find further restrictions on non-abelian -groups acting freely on a product of lens spaces. For actions inducing a trivial action on homology, we reach the following characterization: A -group can act freely on a product of lens spaces with a trivial action on homology if and only if and has the -extension property. The main technique is to study group extensions associated to free actions.

     

The modular group algebra problem for metacyclic -groups

It is shown that the isomorphism type of a metacyclic -group is determined by its group algebra over the field of elements. This completes work of Baginski. It is also shown that, if a -group has a cyclic commutator subgroup , then the order of the largest cyclic subgroup containing is determined by .

     

On the number of conjugacy class sizes and character degrees in finite -groups

In this note we prove that for any two integers 1$"> there exist finite -groups of class such that and .

     

Essential cohomology of finite groups

In this paper we prove that a finite group G with Cohen-Macaulay mod p cohomology will have non-trivial undetectable elements in if and only if G is a p-group such that every element of order p in G is central. Applications and examples are also provided. Received: April 18, 1996     

Exotic cohomology for

We show that for the mod group cohomology of is not detected on diagonal matrices.

     

Nilpotency degree of cohomology rings in characteristic

Let be an odd prime number. The purpose of this paper is to provide a -group whose mod- cohomology ring has a nilpotent element satisfying .

     

Homotopy commutativity of -spaces with finitely generated cohomology

We show that a simply connected homotopy associative and homotopy commutative mod -space with finitely generated mod cohomology is homotopy equivalent to a finite product of , , the three-connected cover and the homotopy fiber of the map for . Our result also shows that a connected -space in the sense of Sugawara with finitely generated mod cohomology has the homotopy type of a finite product of , and for .

     

Cohomology of uniformly powerful -groups

In this paper we will study the cohomology of a family of -groups associated to -Lie algebras. More precisely, we study a category of -groups which will be equivalent to the category of -bracket algebras (Lie algebras minus the Jacobi identity). We then show that for a group in this category, its -cohomology is that of an elementary abelian -group if and only if it is associated to a Lie algebra.

We then proceed to study the exponent of in the case that is associated to a Lie algebra . To do this, we use the Bockstein spectral sequence and derive a formula that gives in terms of the Lie algebra cohomologies of . We then expand some of these results to a wider category of -groups. In particular, we calculate the cohomology of the -groups which are defined to be the kernel of the mod reduction

     

Group homomorphisms inducing cohomology monomorphisms

Let be a homomorphism of -groups such that
is injective, for . We prove that the non-bijectivity of implies the existence of a quotient of containing as a proper direct factor. This gives a refined proof of a result of Evens, which asserts that is bijective if is.

     

Vanishing of the third simplicial cohomology group of

We show that . We first use the Connes-Tzygan exact sequence to prove that this is equivalent to the vanishing of the third cyclic cohomology group , where is the non-unital Banach algebra , and then prove that .

     

Chern classes and extraspecial groups

Summary The mod-p cohomology ring of the extraspecialp-group of exponentp is studied for oddp. We investigate the subquotient ch(G) generated by Chern classes modulo the nilradical. The subring of ch(G) generated by Chern classes of one-dimensional representations was studied by Tezuka and Yagita. The subring generated by the Chern classes of the faithful irreducible representations is a polynomial algebra. We study the interplay between these two families of generators, and obtain some relations between them. The support of the Deutsche Forschungsgemeinschaft and the Max Kade Foundation is gratefully acknowledged, as is the kind hospitality of the ETH Zürich. The support of the Eidgen?ssische Technische Hochschule in Zürich and of the Leibniz Fellowship Programme is acknowledged with thanks. This article was processed by the author using theLatex style filecljour1 from Springer-Verlag.     

Essential laminations in -bundles

We show that, with a few familiar exceptions, every essential lamination in an interval-bundle over a closed surface can be isotoped to lie everywhere transverse to the -fibers of the bundle.

     

-bases for cohomology of local systems on hyperplane complements

We study cohomology with coefficients in a rank one local system on the complement of an arrangement of hyperplanes . The cohomology plays an important role for the theory of generalized hypergeometric functions. We combine several known results to construct explicit bases of logarithmic forms for the only non-vanishing cohomology group, under some nonresonance conditions on the local system, for any arrangement . The bases are determined by a linear ordering of the hyperplanes, and are indexed by certain ``no-broken-circuits" bases of . The basic forms depend on the local system, but any two bases constructed in this way are related by a matrix of integer constants which depend only on the linear orders and not on the local system. In certain special cases we show the existence of bases of monomial logarithmic forms.

     

The cohomology of the Morava stabilizer group at the prime 3

We compute the cohomology of the Morava stabilizer group at the prime by resolving it by a free product and analyzing the ``relation module.'