The ranks of central unit groups of integral group rings of alternating groups

Let G be a finite group and U(Z(Z G)) be the group of units of the center Z(Z G) of the integral group ring Z G (the central unit group of the ring Z G). The purpose of the present work is to study the ranks r _n of groups U(Z(ZAn)), i.e., of central unit groups of integral group rings of alternating groups A _n . We shall find all values n for r _n = 1 and propose an approach on how to describe the groups U(Z(ZAn)) in these cases, and we will present some results of calculations of r _n for n ≤ 600.     

JSJ-splittings for finitely presented groups over slender groups

We generalize the JSJ-splitting of Rips and Sela to give decompositions of finitely presented groups which capture splittings over certain classes of small subgroups. Such classes include the class of all 2-ended groups and the class of all virtually Z⊕Z groups. The approach, called “track zipping”, is relatively elementary, and differs from the Rips-Sela approach in that it does not rely on the theory of R-trees but rather on an understanding of certain embedded 1-complexes (called patterns) in a presentation 2-complex for the ambient group. Oblatum 18-IV-1997 & 30-I-1998 / Published online: 18 September 1998     

Solvable absolute Galois groups are metabelian

We show that solvable absolute Galois groups have an abelian normal subgroup such that the quotient is the direct product of two finite cyclic and a torsion-free procyclic group. In particular, solvable absolute Galois groups are metabelian. Moreover, any field with solvable absolute Galois group G admits a non-trivial henselian valuation, unless each Sylow-subgroup of G is either procyclic or isomorphic to Z ₂⋊Z/2Z. A complete classification of solvable absolute Galois groups (up to isomorphism) is given. Oblatum 22-IV-1998 & 1-IX-2000?Published online: 30 October 2000     

On some ranks of infinite groups

Abstract A group G has finite Hirsch-Zaicev rank r_hz(G) = r if G has an ascending series whose factors are either infinite cyclic or periodic and if the number of infinite cyclic factors is exactly r. The authors discuss groups with finite Hirsch-Zaicev rank and the connection between this and groups having finite section p-rank for some prime p, or p=0. Groups all of whose abelian subgroups are of bounded rank are also discussed. Keywords: p-rank, locally generalized radical group, Hirsch-Zaicev rank, torsion-free rank, rank Mathematics Subject Classification (2000): 20F19, 20E25, 20E15     

Singular Archimedean lattice-ordered groups

W stands for the category of all archimedean l-groups with designated weak unit. The subcategory W _s of all groups with singular weak unit is analyzed as a full subcategory of W which is both epireflective and monocoreflective. A general technique for "contracting" monoreflections of a category A to a monocoreflective subcategory B is developed and then applied to W _s to show that: (i) the projectable hull in W _s is a monoreflection; (ii) essential hulls in W _s are formed by simply taking the lateral completion, and G is essentially closed in this category if and only if , where X is compact, Hausdorff and extremally disconnected; (iii) the maximum monoreflection on W _s , denoted , is obtained by contracting the maximum monoreflection on W, and G is epicomplete in W _s precisely when G is laterally -complete; (iv) the maximum essential reflection on W _s , denoted , is the contraction of the maximum essential reflection on W. Received January 22, 1997; accepted in final form June 11, 1998.     

On sse-embedded subgroups of finite groups

Abstract

In this paper, we introduce the concept of sse-embedded subgroups of finite groups and present some new characterizations of solubility of finite groups using the sse-embedding property of subgroups. Furthermore, we discuss the sse-embedded subgroups in finite nonabelian simple groups. Some previously known results are generalized and unified.     

Universally equivalent solvable groups

Every group that is finitely presented in the varietyA ⁿ of solvable groups. and is universally equivalent to a free group F_r(A ⁿ) in this variety, is embedded in the Cartesian degree of F₂(A ⁿ). All subgroups on a set of two generators in that Cartesian degree which are universally equivalent to F₂(A ⁿ) are determined. Free solvable and nilpotent groups are proved universally equivalent. Supported by RFFR grant No. 99-01-00567, and through the RP “Universities of Russia. Fundamental Research.” Translated fromAlgebra i Logika, Vol. 39, No. 2, pp. 227–240, March–April, 2000.     

Chow groups and Borel-Moore schemes

Summary In this paper we study the Chow groups of schemes for which the class map to Borel-Moore homology is an isomorphism. Then we determine the Chow groups of the scheme Cop^k P ⁿ parametrizing finite coplanary subschemes of lenght k ofP ⁿ and of the variety of «complete S-tuples» of Le Barz.The authors were partially supported by the DGICYT.     

On the Scrimger varieties of lattice-ordered groups

The containmentS _nL _n is proper. (Here n2 is a positive integer,S _n is the small variety of lattice-ordered groups of Scrimger andL _n is the variety of lattice-ordered groups defined by the law aⁿbⁿ=bⁿaⁿ.) Jo E. Smith proved this result for n a composite integer. In this note we proveS _nL _n when n is prime.Presented by L. Fuchs.     

Quasi-varieties of lattice ordered groups

Whereas there is a maximal proper variety of lattice-ordered groups, it is known that there is no maximal proper quasi-variety of lattice-ordered groups. We prove that there are 2º (the maximal possible) pairwise incomparable quasi-varieties of lattice-ordered groups containing . Some of the distributive laws of the semigroup lattice of quasi-varieties are examined and their truth (or falsity) is established. It is also shown here that the latticeL of alll-group varieties is a sublattice of the latticeQ of quasi-varieties ofl-groups but fails to be a complete sublattice.This article is a part of the author's Ph.D. dissertation which was directed by Professor A. M. W. Glass. The author wishes to express his sincere gratitude to Professor Glass for his assistance and encouragement during the writing of the dissertation and this article. He also wishes to thank Professor K. K. Hickin for his help with nilpotent material, in particular for his help in establishing Theorem 4.7.Presented by L. Fuchs.     

Weakly least integer closed groups

An archimedean lattice-ordered groupA with distinguished weak unit has the canonical Yosida representation as an ℓ-group of extended real-valued functions on a certain compact Hausdorff spaceY A. Such an ℓ-groupA is calledleast integer closed, orLIC (resp.,weakly least integer closed, orwLIC) if, in the representation,a ∈A implies [a] ∈A (resp., there isa′ ∈A witha′=[a] on a dense set inY A), where [r] ≡ the least integer greater than or equal tor. Earlier, we have studiedLIC groups, with an emphasis on their a-extensions. Here, we turn towLIC groups: we give an intrinsic (though awk-ward) characterization in terms of existence of certain countable suprema. This results also in an intrinsic characterization ofLIC, previously lacking. Also,wLIC is a hull class (whichLIC is not), and the hullwlA is “somewhere near” the projectable hullpA. The best comparison comes from a (somewhat novel) factoringpA=loc(wpA), wherewpA is the “weakly projectable” hull (defined here), andlocB is the “local monoreflection”; then,wpA≤wlA≤loc(wpA), andpA≤loc(wlA), while with a strong unit, all these coincide. Numerous examples and special cases are examined.     

Limit groups as limits of free groups

We give a topological framework for the study of Sela'slimit groups: limit groups are limits of free groups in a compact space of marked groups. Many results get a natural interpretation in this setting. The class of limit groups is known to coincide with the class of finitely generated fully residually free groups. The topological approach gives some new insight on the relation between fully residually free groups, the universal theory of free groups, ultraproducts and non-standard free groups.     

Finite groups with complemented minimal p-subgroups

Abstract

For a given prime p, we investigate the finite groups in which every minimal p-subgroup is complemented.     

The McKay conjecture for exceptional groups and odd primes

Let G be a simply-connected simple algebraic group over an algebraically closed field of characteristic p with a Frobenius map F : G → G and G := G ^F , such that the root system is of exceptional type or G is a Suzuki group or Steinberg’s triality group. We show that all irreducible characters of C _G (S), the centraliser of S in G, extend to their inertia group in N _G (S), where S is any F-stable Sylow torus of (G, F). Together with the work in [16] this implies that the McKay conjecture is true for G and odd primes ℓ different from the defining characteristic. Moreover it shows important properties of the associated simple groups, which are relevant for the proof that the associated simple groups are good in the sense of Isaacs, Malle and Navarro, as defined in [14]. This research has been supported by the DFG-grant “Die Alperin-McKay-Vermutung für endliche Gruppen” and an Oberwolfach Leibniz fellowship.     

Cohomologically trivial internal categories in categories of groups with operations

We describe cohomologically trivial internal categories in the categoryC of groups with operations satisfying certain conditions ([15], [16]). As particular cases we obtain: ifC=Gr, H⁰(C, –)=0 iff C is a connected internal category; ifC=Ab,H ¹(C, –)=0 iff C is equivalent to the discrete internal category (Cokerd, Cokerd, 1, 1, 1, 1). We also discuss related questions concerning extensions, internal categories, their cohomology and equivalence in the categoryC.     

Finite groups with abelian second maximal subgroups

Abstract

In this article, we give a complete classification of finite groups whose second maximal subgroups are all abelian.     

Variants of Kazhdan’s property for subgroups of semisimple groups

Some variants of Kazhdan’s property (T) for discrete groups are presented. It is shown that some groups (e.g. SL _n (Q),n≧3) which do not have property (T) still have some of these weaker properties. Applications to cohomology and infinitesimal rigidity for certain actions on manifolds are derived. Research partially supported by a grant from the Israel-United States Binational Science Foundation. Research partially supported by NSF Grant.     

On the equations defining irreducible algebraic groups

Abstract

We consider five known problems concerning positive laws in groups. One of them has a counterexample, the others are open in general. It is shown that three of the problems are equivalent and all of them have a positive solution in the class of locally graded groups.     

Finite groups have even more conjugacy classes

In his paper Finite groups have many conjugacy classes (J. London Math. Soc (2) 46 (1992), 239–249), L. Pyber proved the to-date best general lower bounds for the number of conjugacy classes of a finite group in terms of the order of the group. In this paper we strengthen the main results in Pyber’s paper.