Subgroups of powerful groups

We consider subgroups of powerfulp-groups. In particular, we give a new proof that allp-groups are sections of powerfulp-groups, give necessary and sufficient conditions for a 2-generator group to be a normal subgroup of a powerfulp-groups, and show thatp-groups of class 2, orp-groups with a cyclic commutator subgroup, are such normal subgroups.     

K-Theory of curves over number fields

We consider the algebraic K-groups with coefficients of smooth curves over number fields. We give a proof of the Quillen-Lichtenbaum conjecture at the prime 2 and prove explicit corank formulas for the algebraic K-groups with divisible coefficients. At odd primes these formulas assume the Bloch-Kato conjecture, at the prime 2 the formulas hold nonconjecturally.     

Special-valuedl-groups

Special elements and special values have always been of interest in the study of lattice-ordered groups, arising naturally from totally-ordered groups and lexicographic extensions. Much work has been done recently with the class of lattice-ordered groups whose root system of regular subgroups has a plenary subset of special values. We call suchl-groupsspecial- valued. In this paper, we first show that several familiar structures, namely polars, minimal prime subgroups, and the lex kernel, are recognizable from the lattice and the identity. This then leads to an easy proof that special elements can also be recognized from the lattice and the identity. We then give a simple and direct proof thatl, the class of special-valuedl-groups, is closed with respect to joins of convexl-subgroups, incidentally giving a direct proof thatl is a quasitorsion class. This proof is then used to show that the special-valued and finite-valued kernels ofl-groups are recognizable from the lattice and the identity. We also show that the lateral completion of a special-valuedl-group is special-valued and is an a^*-extension of the originall-group.Our most important result is that the lateral completion of a completely distributive normal-valuedl-group is special-valued. This lends itself easily to a new and simple proof of a result by Ball, Conrad, and Darnel that generalizes the Conrad-Harvey-Holland Theorem, namely, that every normal-valuedl-group can be -embedded into a special-valuedl-group.This paper is dedicated to the memory of Prof. Samuel Wolfenstein, who initiated the study of normal-valuedl-groups and recognized early the importance of special-valuedl-groups.Presented by L. Fuchs.     

P.hall’ theorem for regular p-groups and its application to some classification problems

In 1933 P.Hall proved a basis theorem for regular p-groups. In this paper we give another proof for it; this proof allows one to construct a uniqueness basis explicitly. This method to construct uniqueness bases can be used in some classification problems for regular p-groups. Here we give an example, and as a by-product, we obtain a classification for groups of order p ₄ p≥5.     

On the action of the Iwahori–Hecke algebra on modular invariants

We compute the action of the modular Iwahori–Hecke algebra on the ring of invariants of the mod p cohomology of elementary p-groups under Borel subgroup of the general linear group. Applications include a direct proof of the structure of the universal Steenrod algebra and a new proof of a key result on the structure of the Takayasu modules.     

Totient numbers for cyclic group rings

In this note we compute the totient numbers for the rational group-ring of the cyclic groups. We also give a simplified proof of our product formula for the totients and mention some examples for elementary abelian p-groups.     

On a Note From Oliver Concerning Generalized Euclidean Group Rings

In an unpublished 1987 letter, Bob Oliver determined which elementary abelian 2-groups have generalized euclidean integral group rings. He produced a filtration of E ₂(R) by normal subgroups, sandwiched between elementary and special linear relative groups, with layers that are second homology groups with mod-2 coefficients. His proof is presented here, with related consequences for some other finite groups.     

Aglianò-Montagna type decomposition of linear pseudo hoops and its applications

We decompose every linear pseudo hoop as an Aglianò-Montagna type of ordinal sum of linear Wajsberg pseudo hoops which are either negative cones of linear ?-groups or intervals in linear unital ?-groups with strong unit. We apply the decomposition to present a new proof that every linear pseudo BL-algebra and consequently every representable pseudo BL-algebra is good. Moreover, we show that every maximal filter and every value of a linear pseudo hoop is normal, and every σ-complete linear pseudo hoop is commutative.     

Algebraically closed and existentially closed Abelian lattice-ordered groups

If \({\mathcal{G}}\) is an Abelian lattice-ordered (l-) group, then \({\mathcal{G}}\) is algebraically (existentially) closed just in case every finite system of l-group equations (equations and inequations), involving elements of \({\mathcal{G}}\), that is solvable in some Abelian l-group extending \({\mathcal{G}}\) is solvable already in \({\mathcal{G}}\). This paper establishes two systems of axioms for algebraically (existentially) closed Abelian l-groups, one more convenient for modeltheoretic applications and the other, discovered by Weispfenning, more convenient for algebraic applications. Among the model-theoretic applications are quantifierelimination results for various kinds of existential formulas, a new proof of the amalgamation property for Abelian l-groups, Nullstellensätze in Abelian l-groups, and the display of continuum-many elementary-equivalence classes of existentially closed Archimedean l-groups. The algebraic applications include demonstrations that the class of algebraically closed Abelian l-groups is a torsion class closed under arbitrary products, that the class of l-ideals of existentially closed Abelian l-groups is a radical class closed under binary products, and that various classes of existentially closed Abelian l-groups are closed under bounded Boolean products.     

The orbit method for profinite groups and a p-adic analogue of Brown’s theorem

We develop an approach to the character theory of certain classes of finite and profinite groups based on the construction of a Lie algebra associated to such a group, but without making use of the notion of a polarization which is central to the classical orbit method. Instead, Kirillov’s character formula becomes the fundamental object of study. Our results are then used to produce an alternate proof of the orbit method classification of complex irreducible representations of p-groups of nilpotence class < p, where p is a prime, and of continuous complex irreducible representations of uniformly powerful pro-p-groups (with a certain modification for p = 2). As a main application, we give a quick and transparent proof of the p-adic analogue of Brown’s theorem, stating that for a nilpotent Lie group over ℚ_p the Fell topology on the set of isomorphism classes of its irreducible representations coincides with the quotient topology on the set of its coadjoint orbits. The research of M. B. was partially supported by NSF grant DMS-0401164.     

On finite Alperin 2-groups with elementary abelian second commutants

By an Alperin group we mean a group in which the commutant of each 2-generated subgroup is cyclic. Alperin proved that if p is an odd prime then all finite p-groups with this property are metabelian. The today??s actual problem is the construction of examples of nonmetabelian finite Alperin 2-groups. Note that the author had given some examples of finite Alperin 2-groups with second commutants isomorphic to Z ₂ and Z ₄ and proved the existence of finite Alperin 2-groups with cyclic second commutants of however large order by appropriate examples. In this article the existence is proved of finite Alperin 2-groups with abelian second commutants of however large rank.     

A Classification of Regular p-Groups with Invariants (e, 2, 1)

The class of the regular p-groups is one of the important classes in p-groups. Not only it has many similar properties as abelian p-groups, but also many of the p-groups belong to this class. In this paper, using the algorithms for determining the isomorphic regular p-groups, we give a complete classification of the regular p-groups with e-invariants (e, 2, 1).Supported by SXYSF 991003.     

A Descent Theorem in Topological K-Theory

We prove the Lichtenbaum–Quillen conjecture in the topological context: in other words, real K-theory can be deduced from complex K-theory via the usual descent spectral sequence. More precise results are proved, however, and new applications are stated. The main ingredients in the proof are Atiyah's KR-theory and the definition of higher K-groups via Clifford algebras.     

New Classes of Groups Containing Menon Difference Sets

A theorem due to Davis on the existence of Menon difference sets in 2-groups is generalised to non-2-groups. The existence of Menon difference sets in many new non-abelian groups is established.     

Finite 2-groups in which distinct nonlinear irreducible characters have distinct kernels

In this paper, we study finite 2-groups in which distinct nonlinear irreducible characters have distinct kernels. We prove several results concerning these groups and completely classify 2-groups with at most five nonlinear irreducible characters satisfying this property.     

Fibonacci Lengths of Certain Nilpotent 2-Groups

In this paper, we study two classes of 2-generated 2-groups of nilpotency class 2 classified by Kluempen in 2002 and also a class of finite 2-groups of high nilpotency class for their Fibonacci lengths. Their involvement in certain interesting sequences of Tribonacci numbers gives us some explicit formulas for the Fibonacci lengths and this adds to the small class of finite groups for which the Fibonacci length are known.     

Presenting Powers of Augmentation Ideals of Elementary p-Groups

This note presents powers of the augmentation ideal of an integral group ring of an elementary p-group, generalizing results of Bak and Vavilov for elementary 2-groups and of Parmenter for elementary 3-groups.     

Resolvent characterisation of generators of cosine functions and C 0-groups

The paper provides new characterisations of generators of cosine functions and C ₀-groups on UMD spaces and their applications to some classical problems in cosine function theory. In particular, we show that on UMD spaces, generators of cosine functions and C ₀-groups can be characterised by means of a complex inversion formula. This allows us to provide a strikingly elementary proof of Fattorini’s result on square root reduction for cosine function generators on UMD spaces. Moreover, we give a cosine function analogue of McIntosh’s characterisation of the boundedness of the H ^∞ functional calculus for sectorial operators in terms of square function estimates. Another result says that the class of cosine function generators on a Hilbert space is exactly the class of operators which possess a dilation to a multiplication operator on a vector-valued L ₂ space. Finally, we prove a cosine function analogue of the Gomilko-Feng-Shi characterisation of C ₀-semigroup generators and apply it to answer in the affirmative a question by Fattorini on the growth bounds of perturbed cosine functions on Hilbert spaces.     

Ghosts in modular representation theory

A ghost over a finite p-group G is a map between modular representations of G which is invisible in Tate cohomology. Motivated by the failure of the generating hypothesis—the statement that ghosts between finite-dimensional G-representations factor through a projective—we define the ghost number of kG to be the smallest integer l such that the composite of any l ghosts between finite-dimensional G-representations factors through a projective. In this paper we study ghosts and the ghost numbers of p-groups. We begin by showing that a weaker version of the generating hypothesis, where the target of the ghost is fixed to be the trivial representation k, holds for all p-groups. We then compute the ghost numbers of all cyclic p-groups and all abelian 2-groups with C₂ as a summand. We obtain bounds on the ghost numbers for abelian p-groups and for all 2-groups which have a cyclic subgroup of index 2. Using these bounds we determine the finite abelian groups which have ghost number at most 2. Our methods involve techniques from group theory, representation theory, triangulated category theory, and constructions motivated from homotopy theory.     

Determinazione deiT 2-gruppi finiti

Summary A T-group is a group in which normality is transitive, a T₁-group is a group which is not a T-group but all of whose proper subgroups are T-groups, and a T₂-group is a group which is not a T- group or a T₁-group but all of whose proper subgroups are T-groups or T₁-groups. In this note we determine all the finite T₂-groups, and we show that all T₂-groups which are either soluble or 2-groups are finite. We study also the groups in which every proper soluble subgroup is a T-group or a T₁-group.

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Lavoro eseguito nell'ambito del G.N.S.A.G.A. (C.N.R.).