Torsion units in integral group rings of some metabelian groups,II

We prove that any torsion unit of the integral group ring ZG is rationally conjugate to a trivial unit if G = A ⋊ X with both A and X abelian, |Xz.sfnc; < p for every prime p dividing |A| provided either |X| is prime or A ic cyclic.     

Zassenhaus Conjecture (ZC1) on torsion units of integral group rings for some metabelian groups

We prove a conjecture of Zassenhaus that every normalized torsion unit of the integral group ring ZG of a finite group G is rationally conjugate to a group element for some metabelian groups including metacyclic groups G containing a normal cyclic group A such that G/A is cyclic of prime power order. The relative prime case was done in [11]. Received: 21 April 2005     

Applications of the Connes spectrum to C1-dynamical systems,II

We show that for a C¹-dynamical system (A, G, α) with G discrete (abelian) the Connes spectrum Γ(α) is equal to $\text{G}\text{?}$ if and only if every nonzero closed ideal in G × _αA has a nonzero intersection with A. Denote by G_J the closed subgroup of G that leaves fixed the primitive ideal J of A. We show for a general group G that if all isotropy groups G_J are discrete, then GX_αA is simple if and only if A is G-simple and $\text{Γ(}\text{α}\text{) =}\text{G}\text{?}$. This result is applicable not only when G is discrete but also when G?$\text{R}$ or G?$\text{T}$ provided that A is not primitive. Specializing to single automorphisms (i.e., G=$\text{Z}$) we show that if (the transposed of) α acts freely on a dense set of points in $\text{A}\text{?}$, then Λ(α)=$\text{T}$. The converse is only proved when A is of type I.     

Sur la structure galoisienne du groupe des unités d'un corps abélien réel de type (p, p)

Let p be a rational prime. We classify those $\text{Z}$[($\text{Z}$/p$\text{Z}$)²]-modules arising as submodules of the units (mod. torsion) of a real abelian field K with Galois group ($\text{Z}$/p$\text{Z}$)², up to isomorphism and up to genus. Explicit results are given when p is 2 or 3. We apply our classification to discuss the existence of a Minkowski unit in K for arbitrary p.     

Arbitrary torsion classes and almost free abelian groups

Using the set theoretical principle ? for arbitrary large cardinals κ, arbitrary large strongly κ-free abelian groupsA are constructed such that Hom(A, G)={0} for all cotorsion-free groupsG with |G|<κ. This result will be applied to the theory of arbitrary torsion classes for Mod-Z. It allows one, in particular, to prove that the classF of cotorsion-free abelian groups is not cogenerated by aset of abelian groups. This answers a conjecture of Göbel and Wald positively. Furthermore, arbitrary many torsion classes for Mod-Z can be constructed which are not generated or not cogenerated by single abelian groups.     

On the minimum number of disjoint pairs in a family of finite sets

The following conjecture of Katona is proved. Let X be a finite set of cardinality n, 1 ? m ? 2ⁿ. Then there is a family $\text{F}$, |$\text{F}$| = m, such that F ∈ $\text{F}$, G ? X, | G | > | F | implies G ∈ $\text{F}$ and $\text{F}$ minimizes the number of pairs (F₁, F₂), F₁, F₂ ∈ $\text{F}$ F₁ ∩ F₂ = ? over all families consisting of m subsets of X.     

On infinite tournaments with regular automorphism groups

Atournament regular representation (TRR) of an abstract groupG is a tournamentT whose automorphism group is isomorphic toG and is a regular permutation group on the vertices ofT. L. Babai and W. Imrich have shown that every finite group of odd order exceptZ ₃ ×Z ₃ admits a TRR. In the present paper we give several sufficient conditions for an infinite groupG with no element of order 2 to admit a TRR. Among these are the following: (1)G is a cyclic extension byZ of a finitely generated group; (2)G is a cyclic extension byZ _2n+1 of any group admitting a TRR; (3)G is a finitely generated abelian group; (4)G is a countably generated abelian group whose torsion subgroup is finite.     

Complete cohomological functors on groups

If Λ is a ring and A is a Λ-module, then a terminal completion of Ext^1_Λ(A, ) is shown to exist if, and only if, Ext^j_Λ(A, P)=0 for all projective Λ-modules P and all sufficiently large j. Such a terminal completion exists for every A if, and only if, the supremum of the injective lengths of all projective Λ-modules, silp Λ, is finite. Analogous results hold for Ext^1_Λ(,A) and involve spli Λ, the supremum of the projective lengths of the injective Λ-modules. When Λ is an integral group ring $\text{Z}$G, spli$\text{Z}$G is finite implies silp $\text{Z}$G is finite. Also the finiteness of spli is preserved under group extensions. If G is a countable soluble group, the spli $\text{Z}$G is finite if, and only if, the Hirsch number of G is finite.     

On the existence of a free subgroup of finite index

Let G be a finitely generated accessible group. We will study the homology of G with coefficients in the left G-module H¹(G;$\text{Z}$[G]). This G-module may be identified with the G-module of continuous functions with values in $\text{Z}$ on the G-space of ends of G, quotiented by the constant functions. The main result is as follows: Suppose G is infinite, then the abelian group H₁(G;H¹(G;$\text{Z}$[G])) has rank 1 if G has a free subgroup of finite index and it has rank 0 if G has not.     

Polynomial sums over automorphs of a positive definite binary quadratic form

Let P(X) be a homogeneous polynomial in X = (x, y), Q(X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q(X). Let A(m) = {N ∈ $\text{Z}$ × $\text{Z}$ : Q(N) = m}. It is shown that if Σ_N∈A(m)P(N) = 0 for each m = 1, 2, 3,… then Σ_U∈GP(UX) ≡ 0.     

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).      

Asymptotic expansions of matrix coefficients: The real rank one case

Let G be a real reductive Lie group of class $\text{H}$, and suppose that the split rank of G is one. We show that the asymptotic expansions of the Eisenstein integrals given in Harish-Chandra (1) give uniform approximation off of a certain naturally defined compact subset of $\text{A}\text{?}$, the unitary dual of A; G = KAN being an Iwasawa decomposition of G.     

Transitive permutation groups in which all derangements are involutions

Let G be a transitive permutation group in which all derangements are involutions. We prove that G is either an elementary abelian 2-group or is a Frobenius group having an elementary abelian 2-group as kernel. We also consider the analogous problem for abstract groups, and we classify groups G with a proper subgroup H such that every element of G not conjugate to an element of H is an involution.     

When is a Pixley-Roy hyperspace CCC?

A space X is said to satisfy condition (C) if for every Y?X with |Y|=ω₁, any family $\text{G}$ of open subsets of Y with |$\text{G}$|=ω₁ has a countable network. It is easy to see that if X satisfies condition (C), then its Pixley-Roy hyperspace $\text{F}$[X] is CCC. We show that under MA_ω1 condition (C) is also necessary for $\text{F}$[X] to be CCC, but under CH it is not.     

Lifting commuting pairs of C1 algebras

Given a commuting pair $\text{A}$₁, $\text{A}$₂ of abelian $\text{C}{}^1$ subalgebras of the Calkin algebra, we look for a commuting pair $\text{B}$₁,$\text{B}$₂ of $\text{C}{}^1$ subalgebras of $\text{B(H)}$ which project onto $\text{A}$₁ and $\text{A}$₂. We do not insist that $\text{B}$_i, be abelian, so $\text{B}$_i, may contain nontrivial compact operators. If X is the joint spectrum σ($\text{A}$₁, $\text{A}$₂), it is shown that the existence of a pair $\text{B}$₁, $\text{B}$₂ depends only on the element τ in Ext(X) determined by $\text{A}$₁, $\text{A}$₂. The set L(X) of those τ in Ext(X) which “lift” in this sense is shown to be a subgroup of Ext(X) when Ext(X) is Hausdorff, and also when $\text{A}$_i are singly generated. In this latter case, L(X) can be explicitly calculated for large classes of joint spectra. These results are applied to lift certain pairs of commuting elements of the Calkin algebra to pairs of commuting operators.     

On the rational spectra of graphs with abelian singer groups

Let G be a finite abelian group. We investigate those graphs $\text{G}$ admitting G as a sharply 1-transitive automorphism group and all of whose eigenvalues are rational. The study is made via the rational algebra $\text{P}$(G) of rational matrices with rational eigenvalues commuting with the regular matrix representation of G. In comparing the spectra obtainable for graphs in $\text{P}$(G) for various G's, we relate subschemes of a related association scheme, subalgebras of $\text{P}$(G), and the lattice of subgroups of G. One conclusion is that if the order of G is fifth-power-free, any graph with rational eigenvalues admitting G has a cospectral mate admitting the abelian group of the same order with prime-order elementary divisors.     

Fourier-Stieltjes transforms of measures with a certain continuity property

Let G be a compact abelian group whose dual group Γ has a finite torsion subgroup. Let μ?M(G) such that ¦μ ¦assigns no mass to any coset of any closed subgroup of G whose index is infinite. Then there is d > 0, dependent only on ∥ μ ∥, such that if for each $\text{γ ? Γ, ¦}\text{}\text{?}\text{gm(γ)¦ ? 1}\text{or}\text{¦}\text{}\text{?}\text{gm(γ)¦ ? d}$, then the set $\text{{γ: ¦}\text{}\text{?}\text{gm(γ)¦ ? 1}}$ is finite. An upper bound on the cardinality of this set is obtained in terms of ∥ μ ∥and the cardinality of the torsion subgroup of Γ.     

A fundamental region for Hecke's modular group

Hecke proved analytically that when λ ≥ 2 or when $\text{λ = 2}\text{cos}\text{(}\text{π}\text{q}\text{)}$, q ∈ Z, q ≥ 3, then $\text{B(λ) = {τ:}\text{Im}\text{τ > 0, |}\text{Re}\text{τ| <}\text{λ}\text{2}\text{, |τ| > 1}}$ is a fundamental region for the group G(λ) = 〈S_λ, T〉, where S_λ: τ → τ + λ and $\text{T: τ →}\text{?1}\text{τ}$. He also showed that B(λ) fails to be a fundamental region for all other λ > 0 by proving that G(λ) is not discontinuous. We give an elementary proof of these facts and prove a related result concerning the distribution of G(λ)-equivalent points.     

Translation-invariant linear forms on C0(G), C(G), Lp(G) for noncompact groups

Suppose G is a locally compact noncompact group. For abelian such G's, it is shown in this paper that L¹(G), C(G), and L^∞(G) always have discontinuous translation-invariant linear forms(TILF's) while C₀(G) and L^p(G) for 1 < p < ∞ have such forms if and only if $\text{G}\text{H}$ is a torsion group for some open σ-compact subgroup H of $\text{G}$. For σ-compact amenable G's, all the above spaces have discontinuous left TILF's.