Two-knot groups with torsion free abelian normal subgroups of rank two

We show that a torsion free abelian normal subgroup of rank two of a two-knot group which is contained in the commutator subgroup must be free abelian, the centralizer of the abelian subgroup is not contained in the commutator subgroup, and neither of the latter two groups is finitely generated. Furthermore, we characterize algebraically the groups of 2-knots which are cyclic branched covers of twist spun torus knots.     

Formal subgroups of abelian varieties

In this paper, we generalize the result of [12] in the following sense. Let A be an abelian variety over a number field k, let ? be the Néron model of A over the ring of integers O _k of k. Completing ? along its zero section defines a formal group over O _k . We prove that any formal subgroup of the generic fiber of whose closure in is smooth over an open subset of Spec O _k arises in fact from an abelian subvariety of A. The proof is of a transcendental nature and uses the Arakelovian formalism introduced by Bost [3]. Oblatum 2-V-2000 & 28-XI-2000?Published online: 5 March 2001     

Quasi-co-local subgroups of abelian groups

Let {0}≠K be a subgroup of the abelian group G. In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Appl. Math., vol. 249, Chapman & Hall/CRC, Boca Raton, FL, 2006, pp. 29-37], K was called a co-local (cl) subgroup of G if is naturally isomorphic to . We generalize this notion to the quasi-category of abelian groups and call the subgroup K≠{0} of G a quasi-co-local (qcl) subgroup of G if is naturally isomorphic to . We show that qcl subgroups behave quite differently from cl subgroups. For example, while cl subgroups K are pure in G, i.e. G/K is torsion-free if G is torsion-free, any reduced torsion group T can be the torsion subgroup t(G/K) of G/K where G is torsion-free and K is a qcl subgroup of G.     

Characterizing subgroups of compact abelian groups

We prove that every countable subgroup of a compact metrizable abelian group has a characterizing set. As an application, we answer several questions on maximally almost periodic (MAP) groups and give a characterization of the class of (necessarily MAP) abelian topological groups whose Bohr topology has countable pseudocharacter.     

Isolated subgroups of finite abelian groups

Czechoslovak Mathematical Journal - We say that a subgroup H is isolated in a group G if for every x ∈ G we have either x ∈ H or 〈x〉 ∩ H = 1. We describe the set of...     

Co-local subgroups of abelian groups II

In [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] the notion of a co-local subgroup of an abelian group was introduced. A subgroup K of A is called co-local if the natural map is an isomorphism. At the center of attention in [J. Buckner, M. Dugas, Co-local subgroups of abelian groups, in: Abelian Groups, Rings, Modules, and Homological Algebra, in: Lect. Notes Pure and Applied Math., vol. 249, Taylor and Francis/CRC Press, pp. 25-33] were co-local subgroups of torsion-free abelian groups. In the present paper we shift our attention to co-local subgroups K of mixed, non-splitting abelian groups A with torsion subgroup t(A). We will show that any co-local subgroup K is a pure, cotorsion-free subgroup and if D/t(A) is the divisible part of A/t(A)=D/t(A)⊕H/t(A), then K∩D=0, and one may assume that K⊆H. We will construct examples to show that K need not be a co-local subgroup of H. Moreover, we will investigate connections between co-local subgroups of A and A/t(A).     

Basic subgroups in abelian group rings

Suppose is a commutative ring with identity of prime characteristic and is an arbitrary abelian -group. In the present paper, a basic subgroup and a lower basic subgroup of the -component and of the factor-group of the unit group in the modular group algebra are established, in the case when is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed -component and of the quotient group are given when is perfect and is arbitrary whose is -divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring is perfect and is -primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.     

On purifiable subgroups in arbitrary abelian groups

We determine the structure of a p-pure[pure] hull of a p-purifiable [purifiable] subgroup of an arbitrary abelian group. Moreover, we prove that a subgroup A of an abelian group G is purifiable in G if and only if A is p-purifiable in G for every prime p. Using these results, we characterize the groups G for which all subgroups are purifiable in G. Furthermore, we establish several properties of purifiable subgroups.     

Basic subgroups in modular abelian group algebras

Suppose F is a perfect field of char F = p ≠ 0 and G is an arbitrary abelian multiplicative group with a p-basic subgroup B and p-component G _p . Let FG be the group algebra with normed group of all units V(FG) and its Sylow p-subgroup S(FG), and let I _p (FG; B) be the nilradical of the relative augmentation ideal I(FG; B) of FG with respect to B. The main results that motivate this article are that 1 + I _p (FG; B) is basic in S(FG), and B(1 + I _p (FG; B)) is p-basic in V(FG) provided G is p-mixed. These achievements extend in some way a result of N. Nachev (1996) in Houston J. Math. when G is p-primary. Thus the problem of obtaining a (p-)basic subgroup in FG is completely resolved provided that the field F is perfect. Moreover, it is shown that G _p (1 + I _p (FG; B))/G _p is basic in S(FG)/G _p , and G(1 + I _p (FG; B))/G is basic in V(FG)/G provided G is p-mixed. As consequences, S(FG) and S(FG)/G _p are both starred or divisible groups. All of the listed assertions enlarge in a new aspect affirmations established by us in Czechoslovak Math. J. (2002), Math. Bohemica (2004) and Math. Slovaca (2005) as well.     

Intermediately fully invariant subgroups of abelian groups

Describing intermediately fully invariant subgroups of divisible and torsion groups, we show that the intermediately fully invariant subgroups are direct summands in a completely decomposable group whose every homogeneous component is decomposable. For torsion groups, we find out when all their fully invariant subgroups are intermediately fully invariant; and for torsion-free groups, this question comes down to the reduced case. Also, in a torsion group that is the sum of cyclic subgroups, its subgroup is shown to be intermediately inert if and only if it is commensurable with some intermediately fully invariant subgroup.