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We prove that the $p^\infty$ -torsion of the transcendental Brauer group of an abelian variety over a finitely generated field of characteristic $p>0$ is bounded. This answers a (variant of a) question asked by Skorobogatov and Zarhin for abelian varieties. To do this, we prove a 'flat Tate conjecture' for divisors. We also study other geometric Galois-invariant $p^\infty$ -torsion classes of the Brauer group which are not in the transcendental Brauer group. These classes, in contrast with our main theorem, can be infinitely $p$ -divisible. We explain how the existence of these $p$ -divisible towers is naturally related to the failure of surjectivity of specialisation morphisms of Néron–Severi groups in characteristic $p$. [ABSTRACT FROM AUTHOR]

The graphical theory of direct decompositions of torsion-free Abelian groups of a certain type is considered as the basis for the algorithmic construction of the set of their direct decompositions satisfying some specified conditions. [ABSTRACT FROM AUTHOR]

Let k be a number field, p ≥ 2 a prime and S ≠ ∅ a set of finite places of k. We call K / k a totally S -ramified cyclic p -tower if Gal (K / k) ≃ ℤ / p N ℤ and if S is totally ramified. Using analogues of Chevalley's formula [G. Gras, Invariant generalized ideal classes—Structure theorems for p-class groups in p-extensions, Proc. Math. Sci.127(1) (2017) 1–34], we give an elementary proof of a stability theorem (Theorem 3.1) for generalized p -class groups n of the layers k n ⊆ K : let λ = max (0 , # S − 1 − ρ) given in Definition 1.1 (ρ = r 1 (k) + r 2 (k) − 1 for ordinary class groups); then # n = # 0 ⋅ p λ ⋅ n for all n ∈ [ 0 , N ] , if and only if # 1 = # 0 ⋅ p λ . This improves the case λ = 0 of [T. Fukuda, Remarks on ℤ p -extensions of number fields, Proc. Japan Acad. Ser. A70(8) (1994) 264–266; J. Li, Y. Ouyang, Y. Xu and S. Zhang, l-Class groups of fields in Kummer towers, Publ. Mat.  66(1) (2022) 235–267; Y. Mizusawa, K. Yamamoto, On p -class groups of relative cyclic p -extensions, Arch. Math.  117(3) (2021) 253–260] whose techniques are based on Iwasawa's theory or Galois theory of pro- p -groups. We deduce promising capitulation properties of 0 in the tower giving Conjecture 4.1. Finally, we apply our principles to the torsion groups n of abelian p -ramification theory. Numerical examples are given with PARI programs. [ABSTRACT FROM AUTHOR]

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In this paper, we formulate the Drinfeld module analogue of a question raised by Lang and studied by Katz on the existence of rational points on abelian varieties over number fields. Given a maximal ideal of q [ T ] , the question essentially asks whether, up to isogeny, a Drinfeld module ϕ over q (T) contains a rational -torsion point if the reduction of ϕ at almost all primes of q [ T ] contains a rational -torsion point. Similar to the case of abelian varieties, we show that the answer is positive if the rank of the Drinfeld module is 2 , but negative if the rank is 3. Moreover, for rank 3 Drinfeld modules we classify those cases where the answer is positive. [ABSTRACT FROM AUTHOR]

It is a result of Gabriel that hereditary torsion pairs in categories of modules are in bijection with certain filters of ideals of the base ring, called Gabriel filters or Gabriel topologies. A result of Jans shows that this bijection restricts to a correspondence between (Gabriel filters that are uniquely determined by) idempotent ideals and TTF triples. Over the years, these classical results have been extended in several different directions. In this paper, we present a detailed and self-contained exposition of an extension of the above bijective correspondences to additive functor categories over small preadditive categories. In this context, we also show how to deduce parametrizations of hereditary torsion theories of finite type, Abelian recollements by functor categories, and centrally splitting TTFs. [ABSTRACT FROM AUTHOR]

Let 𝔸 [ n ] be the group of n -torsion points of a commutative algebraic group 𝔸 defined over a number field F. For a prime 𝔭 of F , we let N 𝔭 (𝔸 [ n ]) be the number of 𝔽 𝔭 -solutions of the system of polynomial equations defining 𝔸 [ n ] when reduced modulo 𝔭. Here, 𝔽 𝔭 is the residue field at 𝔭. Let π F (x) denote the number of primes 𝔭 of F such that N (𝔭) ≤ x. We then, for algebraic groups of dimension one, compute the k th moment limit M k (𝔸 / F , n) = lim x → ∞ 1 π F (x) ∑ N (𝔭) ≤ x N 𝔭 k (𝔸 [ n ]) by appealing to the Chebotarev density theorem. We further interpret this limit as the number of orbits of the action of the absolute Galois group of F on k copies of 𝔸 [ n ] by an application of Burnside's Lemma. These concrete examples suggest a possible approach for determining the number of orbits of a group acting on k copies of a set. [ABSTRACT FROM AUTHOR]

We determine explicitly the structure of the torsion group over the maximal abelian extension of Q and over the maximal p-cyclotomic extensions of Q for the family of rational elliptic curves given by Y ² = X³ + AX, where A is an integer. [ABSTRACT FROM AUTHOR]

For actions of m commuting endomorphisms of a torsion abelian group we compute the algebraic entropy and the algebraic receptive entropy, showing that the latter one takes finite positive values in many cases when the former one vanishes. [ABSTRACT FROM AUTHOR]

We study Abelian groups A with centrally essential endomorphism ring End A. If A is a such group which is either a torsion group or a non-reduced group, then the ring End A is commutative. We give examples of Abelian torsion-free groups of finite rank with non-commutative centrally essential endomorphism rings. [ABSTRACT FROM AUTHOR]

Let R be a commutative Noetherian ring graded by a torsion-free abelian group G. We introduce the notion of G-graded irreducibility and prove that G-graded irreducibility is equivalent to irreducibility in the usual sense. This is a generalization of Chen and Kim's result in the Z -graded case. We also discuss the concept of the index of reducibility and give an inequality for the indices of reducibility between any radical non-graded ideal and its largest graded subideal. [ABSTRACT FROM AUTHOR]

We consider left-invariant G 2 -structures on 7-dimensional almost Abelian Lie groups. Also, we characterise the associated torsion forms and the full torsion tensor according to the Lie bracket A of the corresponding Lie algebra. In those terms, we establish the algebraic condition on A for each of the possible 16-torsion classes of a G 2 -structure. In particular, we show that four of those torsion classes are not admissible, since τ 3 = 0 implies τ 0 = 0. Finally, we use the above results to provide the algebraic criteria on A , satisfying the harmonic condition div T = 0 , and this allows to identify which torsion classes are harmonic. [ABSTRACT FROM AUTHOR]

Given a compact abelian group G , we study the poset G δ (G) of G δ -dense subgroups of G and the subgroup δ - den (G) : = ⋂ G δ (G) , named G δ -density nucleus of G. It turns out that for every compact abelian group G : (i) the subgroup δ - den (G) is compact metrizable and coincides with the intersection of just two members D 0 , D 1 ∈ G δ (G) ; (ii) if δ - den (G) = { 0 } , then there exists some independent family F in G δ (G) (i.e., with 〈 ⋃ F 〉 = ⨁ i ∈ I D i) such that: (ii 1) | F | = r (G) , the free rank of G , if mG is not metrizable for any positive m ∈ N ; otherwise, (ii 2) when G is (necessarily) bounded torsion, | F | coincides with the least leading Ulm-Kaplansky invariant of G. (iii) in option (ii 1) the members of F can be chosen to be free subgroups of G precisely when G admits a free dense subgroup (equivalently, when r (G) ≥ d (G)); F can have (the maximum possible) size | G | if and only if r (G) = | G | ; (iv) in option (ii 2) the family F can have size | G | if and only if all leading Ulm-Kaplansky invariants of G coincide with | G |. Resuming, the compact abelian groups G with trivial G δ -density nucleus δ - den (G) are, roughly speaking, "maximally fragmented" into G δ -dense subgroups. More precisely, the existence of a pair of subgroups D 0 , D 1 ∈ G δ (G) with D 0 ∩ D 1 = { 0 } yields the existence of a large independent family F = { D i ∈ I } ⊆ G δ (G) with I is as big as possible (i.e., as big as the algebraic structure of G permits in terms of some immediate necessary restraints involving relevant algebraic cardinal invariants of the group, in the spirit of those in (ii)–(iv)). [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. Denote by ((R-mod)op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let T be the subcategory of stable functors. There are two torsion pairs t 1   =   (Gen (− ,   R) ,   T) and t 2   =   (T ,   ℱ 1) , where ℱ1 is the subcategory of ((R-mod)op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly global dimension at most 1, and assume R satisfies that for any exact sequence 0 → M → N → K → 0, if M and N are pure injective, then K is also pure injective. We calculate the cotorsion pair ( ⊥ T ,   ( ⊥ T) ⊥) cogenerated by T clearly. It is shown that G   ∈   ⊥ T if and only if G/t1(G) is a projective object in T , i.e., G/t1(G) = (−,M) for some R-module M; and G   ∈   ( ⊥ T) ⊥ if and only if G/t2(G) is of the form (−, E), where E is an injective R-module. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. Denote by ((R-mod)op, Ab) the category consisting of contravariant functors from the category of finitely presented left R-modules R-mod to the category of abelian groups Ab. An object in ((R-mod)op, Ab) is said to be a stable functor if it vanishes on the regular module R. Let T be the subcategory of stable functors. There are two torsion pairs t 1   =   (Gen (− ,   R) ,   T) and t 2   =   (T ,   ℱ 1) , where ℱ1 is the subcategory of ((R-mod)op, Ab) consisting of functors with flat dimension at most 1. In this article, let R be a ring of weakly global dimension at most 1, and assume R satisfies that for any exact sequence 0 → M → N → K → 0, if M and N are pure injective, then K is also pure injective. We calculate the cotorsion pair ( ⊥ T ,   ( ⊥ T) ⊥) cogenerated by T clearly. It is shown that G   ∈   ⊥ T if and only if G/t1(G) is a projective object in T , i.e., G/t1(G) = (−,M) for some R-module M; and G   ∈   ( ⊥ T) ⊥ if and only if G/t2(G) is of the form (−, E), where E is an injective R-module. [ABSTRACT FROM AUTHOR]

Abstract We prove that, for a finitely generated group hyperbolic relative to virtually abelian subgroups, the generalised word problem for a parabolic subgroup is the language of a real-time Turing machine. Then, for a hyperbolic group, we show that the generalised word problem for a quasiconvex subgroup is a real-time language under either of two additional hypotheses on the subgroup. By extending the Muller–Schupp theorem we show that the generalised word problem for a finitely generated subgroup of a finitely generated virtually free group is context-free. Conversely, we prove that a hyperbolic group must be virtually free if it has a torsion-free quasiconvex subgroup of infinite index with context-free generalised word problem. [ABSTRACT FROM AUTHOR]

We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable Σ3 Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are “computable d-Σ2” (the conjunction of a computable Σ2 sentence and a computable Π2 sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of Q. We show that for some of these groups, the computable Σ3 Scott sentence is best possible, while for others, there is a computable d-Σ2 Scott sentence. [ABSTRACT FROM AUTHOR]

We prove that c.c. torsion abelian groups can be described by a Π 4 0 -predicate that describes the failure of a brute-force diagonalisation attempt on such groups. We show that there is no simpler description since their index set is Π 4 0 -complete. The results can be viewed as a solution to a 60 year-old problem of Mal'cev in the case of torsion abelian groups. We prove that a computable torsion abelian group has one or infinitely many computable copies, up to computable isomorphism. The result confirms a conjecture of Goncharov from the early 1980s for the case of torsion abelian groups. [ABSTRACT FROM AUTHOR]

We study the free Lie ring of rank $2$ in the variety of all centre-by-nilpotent-by-abelian Lie rings of derived length $3$. This is the quotient $L/([\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime })$ with $c\geqslant 2$ where $L$ is the free Lie ring of rank $2$, $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })$ is the $c$th term of the lower central series of the derived ideal $L^{\prime }$ of $L$, and $L^{\prime \prime \prime }$ is the third term of the derived series of $L$. We show that the quotient $\unicode[STIX]{x1D6FE}_{c}(L^{\prime })+L^{\prime \prime \prime }/[\unicode[STIX]{x1D6FE}_{c}(L^{\prime }),L]+L^{\prime \prime \prime }$ is a direct sum of a free abelian group and a torsion group of exponent $c$. We exhibit an explicit generating set for the torsion subgroup. [ABSTRACT FROM AUTHOR]

We prove that for a weakly mixing algebraic action σ : G ↷ ( X , ν ) , the n th cohomology group H n ( G ↷ X ; T ) , after quotienting out the natural subgroup H n ( G , T ) , contains H n ( G , X ˆ ) as a natural subgroup for n = 1 . If we further assume the diagonal actions σ 2 , σ 4 are T -cocycle superrigid and H 2 ( G , X ˆ ) is torsion free as an abelian group, then the above also holds true for n = 2 . Applying it for principal algebraic actions when n = 1 , we show that H 2 ( G , Z G ) is torsion free as an abelian group when G has property (T) as a direct corollary of Sorin Popa's cocycle superrigidity theorem; we also use it (when n = 2 ) to answer, negatively, a question of Sorin Popa on the 2nd cohomology group of Bernoulli shift actions of property (T) groups. [ABSTRACT FROM AUTHOR]

We discuss the decomposability of torsion-free abelian groups. We show that among computable groups of finite rank this property is S30 - complete. However, when we consider groups of infinite rank, it becomes S1¹ - complete, so it cannot be characterized by a first-order formula in the language of arithmetic. [ABSTRACT FROM AUTHOR]

We study (relative) 𝒦-Mittag–Leffler modules, with emphasis on the class 𝒦 of absolutely pure modules. A final goal is to describe the 𝒦-Mittag–Leffler abelian groups as those that are, modulo their torsion part, ℵ1-free. Several more general results of independent interest are derived on the way. In particular, every flat 𝒦-Mittag–Leffler module (for 𝒦 as before) is Mittag–Leffler. A question about the definable subcategories generated by the divisible modules and the torsion-free modules, resp., has been left open. [ABSTRACT FROM PUBLISHER]

LetBbe a torsion-free Abelian group of finite rank withOT(B) = type(ℚ). This paper gives necessary and sufficient conditions for such a groupBto be a finitely faithfulS-groups with the property that every finite rank pure subgroup of a groupAwithExt(A,B) torsion-free isB-cobalanced inA. Several applications of this result are given. In particular, we classify the torsion-free groupsAof finite rank such that Ext(A,B) is torsion-free in the case thatBis an irreducible Murley group. [ABSTRACT FROM AUTHOR]

We define and investigate the class of almost w1-n-simply presented p-torsion abelian groups, which class properly contains the subclasses of almost n-simply presented groups and w1-n-simply presented groups, respectively. The obtained results generalize those obtained by us in Korean J. Math. (2014) and J. Algebra Appl. (2015). [ABSTRACT FROM AUTHOR]

Using the nice properties of thew-divisible weight and thew-divisible groups, we prove a factorization theorem for compact abelian groupsK; namely,K = Ktor × Kd, whereKtoris a bounded torsion compact abelian group andKdis aw-divisible compact abelian group. By Pontryagin duality this result is equivalent to the same factorization for discrete abelian groups proved in [9]. [ABSTRACT FROM AUTHOR]

Abstract: The main aim of this article is to compute all the moments of the number of -torsion elements in some type of finite abelian groups. The averages involved in these moments are those defined for the Cohen–Lenstra heuristics for class groups and their adaptation for Tate–Shafarevich groups. In particular, we prove that the heuristic model for Tate–Shafarevich groups is compatible with the recent conjecture of Poonen and Rains about the moments of the orders of p-Selmer groups of elliptic curves. For our purpose, we are led to define certain polynomials indexed by integer partitions and to study them in a combinatorial way. Moreover, from our probabilistic model, we derive combinatorial identities, some of which appearing to be new, the others being related to the theory of symmetric functions. In some sense, our method therefore gives for these identities a somehow natural algebraic context. [Copyright &y& Elsevier]

It is known that in an Abelian group G that contains no nonzero divisible torsion-free subgroups the intersection of upper nil-radicals of all the rings on G is $\bigcap\limits_{p} pT(G)$ , where T( G) is the torsion part of G. In this work, we define a pure fully invariant subgroup G* ⊇ T( G) of an arbitrary Abelian mixed group G and prove that if G contains no nonzero torsion-free subgroups, then the subgroup $\bigcap\limits_{p} pG^{*}$ is a nil-ideal in any ring on G, and the first Ulm subgroup G1 is its nilpotent ideal. [ABSTRACT FROM AUTHOR]

We obtain the matrix representation of the minimal rational algebra whose subalgebras, up to isomorphism, are quasi-endomorphism rings of strongly indecomposable torsion-free Abelian groups of rank 4 with pseudosocles of rank 1. [ABSTRACT FROM AUTHOR]

A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. An Abelian group is called an RAI-group if there exists a ring on it in which every ideal is absolute. The problem of describing RAI-groups was formulated by L. Fuchs (Problem 93). In this paper, absolute ideals of torsion Abelian groups and torsion Abelian RAI-groups are described. [ABSTRACT FROM AUTHOR]

In this paper, a description of rings of quasi-endomorphisms of almost completely decomposable torsion-free Abelian groups of rank 4 with zero Jacobson radical is obtained. [ABSTRACT FROM AUTHOR]

In this paper, we consider the question of determinability of an Abelian group by its automorphism group in the class of completely decomposable block-rigid torsion-free Abelian groups. [ABSTRACT FROM AUTHOR]

In this paper, we study holomorphically isomorphic Abelian groups, i.e., Abelian groups with isomorphic holomorphs. We also study a generalization of the concept of holomorphic isomorphism, namely, almost holomorphic isomorphism, which is deeply connected with normal Abelian subgroups of holomorphs of Abelian groups. Torsion-free Abelian groups that are determined by their holomorphs are highlighted from different classes. In particular, it has been found that any homogeneous separable group can be determined by its holomorph in the class of all Abelian groups. [ABSTRACT FROM AUTHOR]

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups. [ABSTRACT FROM AUTHOR]

A representation for p-local Abelian torsion-free groups of finite rank is obtained in terms of homomorphisms of p-adic modules of finite rank with fixed basis. [ABSTRACT FROM AUTHOR]

Abstract: For an abstract Abelian group G, denote by the group G endowed with the torsion Bohr topology induced by all (abstract) homomorphisms of G to the torsion subgroup, , of the circle group . Among the main results of this article are: A homomorphism is continuous if and only if . Every compact subset of is countable. A subset is bounded if and only if is compact and countable. Furthermore, if and only if G is finitely generated. Also we prove that is metrizable non-discrete with and is non-metrizable with , where and are the groups of integer and rational numbers, respectively. [Copyright &y& Elsevier]

Abstract: Let G be a finite group and let be the abelian group of equivalence classes of endotrivial kG-modules, where k is an algebraically closed field of characteristic p. We investigate the torsion-free part of the group and look for generators of . We describe three methods for obtaining generators. Each of them only gives partial answers to the question but we obtain more precise results in some specific cases. We also conjecture that can be generated by modules belonging to the principal block and we prove the conjecture in some cases. [Copyright &y& Elsevier]

Zassenhaus Conjecture for torsion units states that every augmentation 1 torsion unit of the integral group ring of a finite group G is conjugate to an element of G in the units of the rational group algebra ℚ G. This conjecture has been proved for nilpotent groups, metacyclic groups and some other families of groups. It has been also proved for some special groups. We prove the conjecture for cyclic-by-abelian groups. [ABSTRACT FROM AUTHOR]

We prove that a partially commutative metabelian group is a subgroup in a direct product of torsion-free abelian groups and metabelian products of torsion-free abelian groups. From this we deduce that all partially commutative metabelian (nonabelian) groups generate the same quasivariety and prevariety. On the contrary, there exists an infinite chain of different quasivarieties generated by partially commutative groups with defining graphs of diameter 2. [ABSTRACT FROM AUTHOR]

The Pinsker subgroup of an abelian group with respect to an endomorphism was introduced in the context of algebraic entropy. Motivated by the nice properties and characterizations of the Pinsker subgroup, we generalize its construction in two directions. Indeed, we introduce the concept of entropy function h of an abelian category, and we define the Pinsker radical with respect to h, so that the class of all objects with trivial Pinsker radical is the torsion-free class of a torsion theory. [ABSTRACT FROM AUTHOR]

Let $\Bbb A$ be a commutative algebraic group defined over a number field K. For a prime ℘ in K where $\Bbb A$ has good reduction, let N℘,n be the number of n-torsion points of the reduction of $\Bbb A$ modulo ℘ where n is a positive integer. When $\Bbb A$ is of dimension one and n is relatively prime to a fixed finite set of primes depending on $\Bbb A_(/K)$, we determine the average values of N℘,n as the prime ℘ varies. This average value as a function of n always agrees with a divisor function. [ABSTRACT FROM AUTHOR]

In this paper, the classification of rings of quasi-endomorphisms of strongly indecomposable Abelian torsion-free groups of rank 4 with pseudosocles of rank 3 is obtained. [ABSTRACT FROM AUTHOR]

The reduced unitary Whitehead group SK1 of a graded division algebra equipped with a unitary involution (that is, an involution of the second kind) and graded by a torsion-free abelian group is studied. It is shown that calculations in the graded setting are much simpler than their non-graded counterparts. The bridge to the non-graded case is established by proving that the unitary SK1 of a tame-valued division algebra with a unitary involution over a henselian field coincides with the unitary SK1 of its associated graded division algebra. As a consequence, the graded approach allows us not only to recover results available in the literature with substantially easier proofs, but also to calculate the unitary SK1 for much wider classes of division algebras over henselian fields. [ABSTRACT FROM AUTHOR]

A group is called capable if it is a central factor group. In this paper, we establish a necessary condition for a finitely generated non-torsion group of nilpotency class 2 to be capable. Using the classification of two-generator non-torsion groups of nilpotency class 2, we determine which of them are capable and which are not and give a necessary and sufficient condition for a two-generator non-torsion group of class 2 to be capable in terms of the torsion-free rank of its factor commutator group. [ABSTRACT FROM PUBLISHER]

Abstract: Let B be a torsion-free Abelian group of finite rank with . This paper gives necessary and sufficient conditions for such a group B to be a finitely faithful S-group with the property that every finite rank pure subgroup of a group A with torsion-free is B-cobalanced in A. Several applications of this result are given. In particular, we classify the torsion-free groups A of finite rank such that is torsion-free in the case that B is an irreducible Murley group. [Copyright &y& Elsevier]

Abstract: Suppose that and that S, T are sets of primes. Then the classification problem for the S-local torsion-free abelian groups of rank n is Borel reducible to the classification problem for the T-local torsion-free abelian groups of rank n if and only if . [ABSTRACT FROM AUTHOR]

Let B be a torsion-free finitely faithful S-group. This paper investigates the class of all torsion-free Abelian groups A with the property that Ext(A, B) is torsion-free. Our discussion is based on a natural extension of the concept of coseparability. [ABSTRACT FROM AUTHOR]

Cochran–Orr–Teichner introduced in [‘Knot concordance, Whitney towers and L2-signatures’, Ann. of Math. (2) 157 (2003) 433–519] a natural filtration of the smooth knot concordance group called the (n)-solvable filtration. We show that each associated graded abelian group , n ≥ 2, contains infinite linearly independent sets of elements of order 2 (this was known previously for n = 0, 1). Each of the representative knots is negative amphichiral, with vanishing s-invariant, τ-invariant, δ-invariants and Casson–Gordon invariants. Moreover, each is slice in a rational homology 4-ball. In fact we show that there are many distinct such classes in , distinguished by their Alexander polynomials and, more generally, by the torsion in their higher-order Alexander modules. [ABSTRACT FROM AUTHOR]

Abstract: The object of this article are torsion subgroups of the normalized unit group of the integral group ring of a finite group G. For specific subgroups W we study the Gruenberg–Kegel graph . It is shown that the central elements of an isolated subgroup U of a group basis H of are the normalized units of its centralizer ring . Moreover . If G has elementary abelian Sylow 2-subgroups of order at most 8 each finite 2-subgroup of is rationally conjugate to a subgroup of G. Finally torsion subgroups of in the case when G is a minimal simple group are considered. It follows that if G is a simple group which admits a non-trivial partition for each prime p the p-rank of a torsion subgroup of is bounded by that one of G. [ABSTRACT FROM AUTHOR]

Abstract: We give new examples of FA presentable torsion-free abelian groups. Namely, for every , we construct a rank n indecomposable torsion-free abelian group which has an FA presentation. We also construct an FA presentation of the group in which every nontrivial cyclic subgroup is not FA recognizable. [Copyright &y& Elsevier]