Countably valued lattice-ordered groups

A countably valued lattice-ordered group is a lattice-ordered group in which every element has only countably many values. Such lattice-ordered groups are proven to be normal-valued and, though not necessarily special-valued, every element in a countably valuedl-group must have a special value. The class of countably valuedl-groups forms a torsion class, and the torsion radical determined by this class is anl-ideal that is the intersection of all maximal countably valued subgroups.Countably valuedl-groups are shown to be closed with respect toeventually constant sequence extensions, and it is shown that many properties of anl-group pass naturally to its eventually constant sequence extension.Presented by M. Henriksen.     

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.     

On Units of Group Algebras of 2-Groups of Maximal Class

Abstract

We determine the number of elements of order two in the group of normalized units V(𝔽₂ G) of the group algebra 𝔽₂ G of a 2-group of maximal class over the field 𝔽₂ of two elements. As a consequence for the 2-groups G and H of maximal class we have that V(𝔽₂ G) and V(𝔽₂ H) are isomorphic if and only if G and H are isomorphic.     

Categorified central extensions, étale Lie 2-groups and Lie?s Third Theorem for locally exponential Lie algebras

Lie?s Third Theorem, asserting that each finite-dimensional Lie algebra is the Lie algebra of a Lie group, fails in infinite dimensions. The modern account on this phenomenon is the integration problem for central extensions of infinite-dimensional Lie algebras, which in turn is phrased in terms of an integration procedure for Lie algebra cocycles.This paper remedies the obstructions for integrating cocycles and central extensions from Lie algebras to Lie groups by generalising the integrating objects. Those objects obey the maximal coherence that one can expect. Moreover, we show that they are the universal ones for the integration problem.The main application of this result is that a Mackey-complete locally exponential Lie algebra (e.g., a Banach–Lie algebra) integrates to a Lie 2-group in the sense that there is a natural Lie functor from certain Lie 2-groups to Lie algebras, sending the integrating Lie 2-group to an isomorphic Lie algebra.     

Covers of the Abelian Variety of Generalized MV-Algebras

Using the categorical equivalence of the class of generalized MV-algebras with the class of unital ?-groups, we describe all varieties of symmetric top abelian unital ?-groups that cover the variety  _u? of abelian unital ?-groups. Equivalently, we describe all cover varieties of the variety of MV-algebras, ?, within the variety of generalized MV-algebras admitting only one negation and each of whose maximal ideals is normal. In particular, there are continuum many representable varieties of generalized MV-algebras that cover ?.     

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.     

Minimal Non-FO-Groups

A group G is said to be a “minimal non-FO-group” (an MNFO-group) if all its proper subgroups are FO-groups, but G itself is not. The aim of this article is to study the class of MNFO-groups. The structure of MNFO-groups is completely described, both in nonperfect case and perfect case.     

A generalization of 2-Baer groups

A group in which all cyclic subgroups are 2-subnormal is called a 2-Baer group. The topic of this paper are generalized 2-Baer groups, i.e., groups in which the non-2-subnormal cyclic subgroups generate a proper subgroup of the group. If this subgroup is non-trivial, the group is called a generalized T₂-group. In particular, we provide structure results for such groups, investigate their nilpotency class, and construct examples of finite p-groups which are generalized T₂-groups.     

On the Chermak–Delgado lattice of a finite group

Abstract

By imposing conditions upon the index of a self-centralizing subgroup of a group, and upon the index of the center of the group, we are able to classify the Chermak-Delgado lattice of the group. This is our main result. We use this result to classify the Chermak-Delgado lattices of dicyclic groups and of metabelian p-groups of maximal class.     

On maximal abelian subgroups in finite 2-groups

We determine here up to isomorphism the structure of any finite nonabelian 2-group G in which every two distinct maximal abelian subgroups have cyclic intersection. We obtain five infinite classes of such 2-groups (Theorem 1.1). This solves for p = 2 the problem Nr. 521 stated by Berkovich (in preparation). The more general problem Nr. 258 stated by Berkovich (in preparation) about the structure of finite nonabelian p-groups G such that A ∩ B = Z(G) for every two distinct maximal abelian subgroups A and B is treated in Theorems 3.1 and 3.2. In Corollary 3.3 we get a new result for an arbitrary finite 2-group. As an application of Theorems 3.1 and 3.2, we solve for p = 2 a problem of Heineken-Mann (Problem Nr. 169 stated in Berkovich, in preparation), classifying finite 2-groups G such that A/Z(G) is cyclic for each maximal abelian subgroup A (Theorem 4.1).      

On the algebraicK-theory of twisted flag varieties

The algebraicK-groups of projective homogeneous varieties are computed. The answer is given in terms ofK-groups of a semisimple algebra canonically associated with the variety. Our results generalize a result of Quillen and a result of Swan, whereK-groups of Severi-Brauer varieties and of smooth projective quadratic hypersurfaces were computed.     

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.     

Regular CW-Complexes and Poset Resolutions of Monomial Ideals

Abstract

We study the generation of a finite group by its conjugacy classes, while generalizing basic concepts from linear algebra: basis and dimension. Besides the well known Burnside Basis Theorem for finite p-groups, there is no direct extension of these concepts to other families of finite groups. We show that by considering generating sets consisting of conjugacy classes, there is a possibility for such a generalization.     

Groupable lattices

A lattice is called groupable provided it can be endowed with the structure of an l-group (lattice ordered group). The primary objective of this paper is to introduce an order theoretic property of groupable lattices which implies that all associated l-groups are subdirect products of totally ordered groups. This is an analog to Iwasawa's well-known result which asserts that a conditionally complete l-group is abelian. A secondary objective is to outline a general method for identifying classes of l-groups determined by order theoretic properties.     

On <Emphasis Type="Italic">T</Emphasis>-groups,supersolvable groups,and maximal subgroups

A group is called a T-group if all its subnormal subgroups are normal. Finite T-groups have been widely studied since the seminal paper of Gaschütz (J. Reine Angew. Math. 198 (1957), 87–92), in which he described the structure of finite solvable T-groups. We call a finite group G an NNM-group if each non-normal subgroup of G is contained in a non-normal maximal subgroup of G. Let G be a finite group. Using the concept of NNM-groups, we give a necessary and sufficient condition for G to be a solvable T-group (Theorem 1), and sufficient conditions for G to be supersolvable (Theorems 5, 7 and Corollary 6).     

ON MINIMAL NON-<Emphasis Type="Italic">MSP</Emphasis>-GROUPS

A finite group G is called an MSP-group if all maximal subgroups of the Sylow subgroups of G are S-quasinormal in G: We give a complete classification of groups that are not MSP-groups but all their proper subgroups are MSP-groups.     

Finite Soluble Groups with Permutable Subnormal Subgroups

A finite group G is said to be a PST-group if every subnormal subgroup of G permutes with every Sylow subgroup of G. We shall discuss the normal structure of soluble PST-groups, mainly defining a local version of this concept. A deep study of the local structure turns out to be crucial for obtaining information about the global property. Moreover, a new approach to soluble PT-groups, i.e., soluble groups in which permutability is a transitive relation, follows naturally from our vision of PST-groups. Our techniques and results provide a unified point of view for T-groups, PT-groups, and PST-groups in the soluble universe, showing that the difference between these classes is quite simply their Sylow structure.     

A Classification of Minimal Non-MNP-Groups

A finite group G is called an MNP-group if all maximal subgroups of every Sylow subgroup of G are normal in G. In this article, we give a complete classification of those groups which are not MNP-groups but all of whose proper subgroups are MNP-groups.