Cellular covers of totally ordered abelian groups

Cellular covers of groups, and in particular, those of divisible abelian groups, were studied in [FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of groups, J. Pure Appl. Algebra 208, (2007), 61–76], [CHA-CHÓLSKI, W.—FARJOUN, E. D.—GÖBEL, R.—SEGEV, Y.: Cellular covers of divisible abelian groups. In: Contemp. Math. 504, Amer. Math. Soc., Providence, RI, 2009, pp. 77–97], and continued in [FUCHS, L.—GÖBEL, R.: Cellular covers of abelian groups, Results Math. 53, (2009), 59–76] for abelian groups in general. In this note we are investigating cellular covers in the category of totally ordered abelian groups (called o-cellular covers; for definition see Section 2). Some results are similar to those on torsion-free abelian groups (unordered), while others are completely different. For instance, though kernels of o-cellular covers can not be non-zero divisible groups (Lemma 3.1), they may contain non-zero divisible subgroups (Example 3.2); however, the divisible part can not be much larger than the reduced part (Theorem 3.4). There are o-groups, even among the additive subgroups of the rationals, whose o-cellular covers form a proper class (Theorem 4.3).     

Stably free cancellation for abelian group rings

We show that if Γ is a finitely generated abelian group, then every stably free module over Z[Γ] is free.     

On diversity and stability of unit bases for the Euclidean metric

We prove the existence of a family Ω(n) of 2 ^c (where c is the cardinality of the continuum) subgraphs of the unit distance graph (E ⁿ , 1) of the Euclidean space E ⁿ , n ≥ 2, such that (a) for each graph G ? Ω(n), any homomorphism of G to (E ⁿ , 1) is an isometry of E ⁿ ; moreover, for each subgraph G ₀ of the graph G obtained from G by deleting less than c vertices, less than c stars, and less than c edges (we call such a subgraph reduced), any homomorphism of G ₀ to (E ⁿ , 1) is an isometry (of the set of the vertices of G ₀); (b) each graph G ? Ω(n) cannot be homomorphically mapped to any other graph of the family Ω(n), and the same is true for each reduced subgraph of G.     

Regular principal factors in free objects in varieties in the interval [B 2 ,NB 2 ∨G n ]

The author has shown previously how to associate a completely 0-simple semigroup with a connected bipartite graph containing labelled edges and how to describe the regular principal factors in the free objects in the Rees-Sushkevich varieties RS _n generated by all completely 0-simple semigroups over groups from the Burnside variety G _n of groups of exponent dividing a positive integer n by employing this graphical construction. Here we consider the analogous problem for varieties containing the variety B ₂ , generated by the five element Brandt semigroup B ₂, and contained in the variety NB ₂ ∨G _n where NB ₂ is the variety generated by all left and right zero semigroups together with B ₂. The interval [NB ₂ ,NB ₂ ∨G _n ] is of particular interest as it is an important interval, consisting entirely of varieties generated by completely 0-simple semigroups, in the lattice of subvarieties of RS _n .     

Primitive words,free factors and measure preservation

Let F _k be the free group on k generators. A word w ∈ F _k is called primitive if it belongs to some basis of F _k . We investigate two criteria for primitivity, and consider more generally subgroups of F _k which are free factors. The first criterion is graph-theoretic and uses Stallings core graphs: given subgroups of finite rank H ≤ J ≤ F _k we present a simple procedure to determine whether H is a free factor of J. This yields, in particular, a procedure to determine whether a given element in F _k is primitive. Again let w ∈ F _k and consider the word map w: G × … × G → G (from the direct product of k copies of G to G), where G is an arbitrary finite group. We call w measure preserving if given uniform measure on G × … × G, w induces uniform measure on G (for every finite G). This is the second criterion we investigate: it is not hard to see that primitivity implies measure preservation, and it was conjectured that the two properties are equivalent. Our combinatorial approach to primitivity allows us to make progress on this problem and, in particular, prove the conjecture for k = 2. It was asked whether the primitive elements of F _k form a closed set in the profinite topology of free groups. Our results provide a positive answer for F ₂.     

On the universal coefficients formula for shape homology

In this paper it is investigated whether various shape homology theories satisfy the Universal Coefficients Formula (UCF). It is proved that pro-homology and strong homology satisfy UCF in the class FAB of finitely generated abelian groups, while they do not satisfy UCF in the class AB of all abelian groups. Two new shape homology theories (called UCF-balanced) are constructed. It is proved that balanced pro-homology satisfies UCF in the class AB, while balanced strong homology satisfies UCF only in the class FAB.     

When is the radical associated with a module a torsion?

For an arbitrary R-module M we consider the radical (in the sense of Maranda)G _M, namely, the largest radical among all radicalsG, such thatG(M). We determine necessary and sufficient on M in order for the radicalG(M) to be a torsion. In particular,G(M) is a torsion if and only if M is a pseudo-injective module.     

Sylow-metacyclic groups andQ-admissibility

A finite groupG isQ-admissible if there exists a division algebra finite dimensional and central overQ which is a crossed product forG. AQ-admissible group is necessarily Sylow-metacyclic (all its Sylow subgroups are metacyclic). By means of an investigation into the structure of Sylow-metacyclic groups, the inverse problem (is every Sylow-metacyclic groupQ-admissible?) is essentially reduced to groups of order 2 ^a 3 ^b and to a list of known “almost simple” groups.     

Shintani lifting and real-valued characters

We study Shintani lifting of real-valued irreducible characters of finite reductive groups. In particular, if G is a connected reductive group defined over ${\mathbb{F}_q}$ , and ψ is an irreducible character of G( ${\mathbb{F}_{q^m}}$ ) which is the lift of an irreducible character χ of G( ${\mathbb{F}_q}$ ), we prove ψ is real-valued if and only if χ is real-valued. In the case m = 2, we show that if χ is invariant under the twisting operator of G( ${\mathbb{F}_{q^2}}$ ), and is a real-valued irreducible character in the image of lifting from G( ${\mathbb{F}_q}$ ), then χ must be an orthogonal character. We also study properties of the Frobenius–Schur indicator under Shintani lifting of regular, semisimple, and irreducible Deligne–Lusztig characters of finite reductive groups.     

Modules over group rings of solvable groups with rank restrictions on subgroups

Let A be an R G-module over a commutative ring R, where G is a group of infinite section p-rank (0-rank), C _G (A) = 1, A is not a Noetherian R-module, and the quotient A/C _A (H) is a Noetherian R-module for every proper subgroup H of infinite section p-rank (0-rank). We describe the structure of solvable groups G of this type.     

Conway’s Question: The Chase for Completeness

We study various degrees of completeness for a Tychonoff space X. One of them plays a central role, namely X is called a Conway space if X is sequentially closed in its Stone–?ech compactification β X (a prominent example of Conway spaces is provided by Dieudonné complete spaces). The Conway spaces constitute a bireflective subcategory Conw of the category Tych of Tychonoff spaces. Replacing sequential closure by the general notion of a closure operator C, we introduce analogously the subcategory Conw _C of C-Conway spaces, that turns out to be again a bireflective subcategory of Tych. We show that every bireflective subcategory of Tych can be presented in this way by building a Galois connection between bireflective subcategories of Tych and closure operators of Top finer than the Kuratowski closure. Other levels of completeness are considered for the (underlying topological spaces of) topological groups. A topological group G is sequentially complete if it is sequentially closed in its Ra?kov completion ${ \ifmmode\expandafter\tilde\else\expandafter\~\fi{G}}$ . The sequential completeness for topological groups is stronger than Conway’s property, although they coincide in some classes of topological groups, for example: free (Abelian) topological groups, pseudocompact groups, etc.     

Real quadrics in C n , complex manifolds and convex polytopes

In this paper, we investigate the topology of a class of non-Kähler compact complex manifolds generalizing that of Hopf and Calabi-Eckmann manifolds. These manifolds are diffeomorphic to special systems of real quadrics C ⁿ which are invariant with respect to the natural action of the real torus (S ¹) ⁿ onto C ⁿ . The quotient space is a simple convex polytope. The problem reduces thus to the study of the topology of certain real algebraic sets and can be handled using combinatorial results on convex polytopes. We prove that the homology groups of these compact complex manifolds can have arbitrary amount of torsion so that their topology is extremely rich. We also resolve an associated wall-crossing problem by introducing holomorphic equivariant elementary surgeries related to some transformations of the simple convex polytope. Finally, as a nice consequence, we obtain that affine non-Kähler compact complex manifolds can have arbitrary amount of torsion in their homology groups, contrasting with the Kähler situation.     

Følner sequences and Hilbert’s Irreducibility Theorem over Q

Letf(X; T ₁, ...,T _n) be an irreducible polynomial overQ. LetB be the set ofb teZ ⁿ such thatf(X;b) is of lesser degree or reducible overQ. Let ?={F _j}{F _j } _j?1 ^∞ be a Følner sequence inZ ⁿ — that is, a sequence of finite nonempty subsetsF _j ?Z ⁿ such that for eachvteZ ⁿ , $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap (F_j + \upsilon )} \right|}}{{\left| {F_j } \right|}} = 1$ Suppose ? satisfies the extra condition that forW a properQ-subvariety ofP ⁿ ?A ⁿ and ?>0, there is a neighborhoodU ofW(R) in the real topology such that $\mathop {lim sup}\limits_{j \to \infty } \frac{{\left| {F_j \cap U} \right|}}{{\left| {F_j } \right|}}< \varepsilon $ whereZ ⁿ is identified withA ⁿ (Z). We prove $\mathop {lim}\limits_{j \to \infty } \frac{{\left| {F_j \cap B} \right|}}{{\left| {F_j } \right|}} = 0$ .     

Applications of a theorem by Ky Fan in the theory of graph energy

The energy of a graph G is equal to the sum of the absolute values of the eigenvalues of G, which in turn is equal to the sum of the singular values of the adjacency matrix of G. Let X, Y, and Z be matrices, such that X+Y=Z. The Ky Fan theorem establishes an inequality between the sum of the singular values of Z and the sum of the sum of the singular values of X and Y. This theorem is applied in the theory of graph energy, resulting in several new inequalities, as well as new proofs of some earlier known inequalities.     

A note on cosubnormal subgroups

Two subgroupsH andK of a groupG are called cosubnormal if they are both subnormal in the subgroup generated by them. In this paper some subnormality criteria for cosubnormal subgroups of nilpotent-by-abelian groups are given.     

Boolean topological graphs of semigroups: the lack of first-order axiomatization

The graph of an algebra A is the relational structure G(A) in which the relations are the graphs of the basic operations of A. For a class ?? of algebras let G(??)={G(A)∣A∈??}. Assume that ?? is a class of semigroups possessing a nontrivial member with a neutral element and let ? be the universal Horn class generated by G(??). We prove that the Boolean core of ?, i.e., the topological prevariety generated by finite members of ? equipped with the discrete topology, does not admit a first-order axiomatization relative to the class of all Boolean topological structures in the language of ?. We derive analogous results when ?? is a class of monoids or groups with a nontrivial member.     

On reconstructing separable reducedp-groups with a given socle

Let \(\bar B^* \) be a separable reduced (abelian)p-group which is torsion complete. We ask whether for \(G \subseteq \bar B^* \) there is \(H \subseteq _{pr} \bar B^* ,H[p] = G[p]\) ,H[p]=G[p],H not isomorphic toG. IfG is the sum of cyclic groups or is torsion complete, the answer is easily no. For otherG, we prove that the answer is yes assuming G.C.H. Even without G.C.H. the answer is yes if the density character ofG is equal to Min _n<ω|p ⁿG|, i.e., $$\mathop {Min}\limits_{n< \omega } |p^n G| = \mathop {Min}\limits_m \mathop \Sigma \limits_{n > m} |(p^n G)[p]/(p^{n + 1} G)[p]|$$ Of course, instead of two non-isomorphic we can get many, but we do not deal much with this.