P-local Groups Research Articles

Indecomposable p-Local Torsion-Free Groups with Quadratic and Cubic Splitting Fields

Indecomposable torsion-free p-local Abelian groups of finite rank with quadratic and cubic splitting field K are characterized. As a consequence, for groups with quadratic splitting field K it is proved that K-decomposable p-local torsion-free Abelian groups of finite rank are isomorphic if and only if their endomorphism rings are isomorphic.

Partial Decomposition Bases and Global Warfield Groups

The class of abelian groups with partial decomposition bases was developed by the first author in order to generalize Barwise and Eklof's classification of torsion groups in L∞ω. In this article, we continue to explore algebraic characteristics of this class and establish a uniqueness theorem, extending our previous work on mixed p-local groups to the global case. It is shown that groups with partial decomposition bases are characterized in terms of Warfield groups and k-groups of Hill and Megibben. In fact, we prove that the class of groups with partial decomposition bases is identical to the class of k-groups, and, as such, closed under direct summands, and that every finitely generated subgroup of a k-group is locally nice. Also, we introduce and explore subgroups possessing a partial subbasis. As an application, it is shown that isotype k-subgroups of abelian groups are k-groups.

On asymptotic dimension with linear control

We construct a countable p-local group with a proper invariant metric whose Assouad–Nagata dimension is strictly greater than the asymptotic dimension with linear control. This solves Problem 8.6 from the list [4].We study asymptotic dimension with linear control ℓ-asdimω that depends on a fixed ultrafilter ω on N. It turns out that the asymptotic Assouad–Nagata dimension is the supremum of ℓ-asdimω over all ω and the asymptotic dimension with linear control is the minimum of ℓ-asdimω over all ω.

A group-theoretic approach to a family of 2-local finite groups constructed by Levi and Oliver

We extend the notion of a p-local finite group (defined in [BLO03]) to the notion of a p-local group. We define morphisms of p-local groups, obtaining thereby a category, and we introduce the notion of a representation of a p-local group via signalizer functors associated with groups. We construct a chain G = (G_0 → G_1 → ...) of 2-local finite groups, via a representation of a chain G^* = (G_0 → G_1 → ...) of groups, such that G_0 is the 2-local finite group of the third Conway sporadic group Co_3, and for n > 0, G_n is one of the 2-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit G of G exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain G^*. The 2-completed classifying space of G is shown to be the classifying space B DI(4) of the exotic 2-compact group of Dwyer and Wilkerson [DW93].

LOCALIZING GROUPS WITH ACTION

When localizing the semidirect product of two groups, the effect on the factors is made explicit. As an application in Topology, we show that the loop space of a based connected CW-complex is a P-local group, up to homotopy, if and only if p1X and the free homotopy groups [Sk-1, OX], k = 2, are P-local.

THE CLASSIFICATION PROBLEM FOR p-LOCAL TORSION-FREE ABELIAN GROUPS OF RANK TWO

We prove that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility.

ISOTYPE SUBGROUPS OF LOCAL WARFIELD GROUPS

Necessary and sufficient conditions are given for an isotype subgroup H of a p-local Warfield group G to be, itself, Warfield.

P-localization of relative groups

A relative group is a pair ( N, E) consisting of a group E and a normal subgroup N of E. Localization at sets of primes with respect to any radical is introduced as a functor on the category of relative groups. Given a radical R and a set of primes P, we consider the full subcategory of relative groups ( N, E) which are P-local with respect to R, i.e. N is a P-local group and R( E/ N) is trivial. We construct a left adjoint to the inclusion of this subcategory into the category of relative groups. As an application, when the radical R assigns the P′-isolator or the P′-torsion subgroup to a group, the effect of the classical P-localization is made explicit for groups fitting into an extension with a quotient F such that F/ RF is torsion.

The duals of Warfield groups

All considered groups in this paper will be abelian groups. By a p-local group we mean a module over Zp, the ring of integers localized at the prime p. Recall that an abelian group is said to be simply presented provided it can be defined in terms of generators and relations in such a way that all of the relations are of the form mx = 0 or mx = y. For instance, totally projective p-groups and completely decomposable torsion-free groups are simply presented. Warfield groups are direct summands of simply presented groups, in the p-local case they are also called Warfield modules. Warfield [W] established some characterizations of Warfield modules and gave a complete set of isomorphism invariants. Using an alternate definition, Hunter, Richman and Walker [HRW1], [HRW2], [HRi] proved existence theorems for p-local and global Warfield groups and were able to classify global Warfield groups. Moore [M] described Warfield modules in terms of quasi-sequentially nice submodules. Introducing the concept of a knice submodule, Hill and Megibben [HM1] characterized Warfield modules using the third axiom of countability. In [HM3], they proved a variety of characterizations of Warfield groups using Axiom 3 and the global definitions of nice and knice subgroups. Some more characterizations of Warfield groups are contained in [L1]. In [F], the following question was formulated as Problem 65: Which are the compact abelian groups whose duals are totally projective p-groups? Kiefer [K] described those groups dualizing the various characterizations of totally projective p-groups. Since Warfield groups are generalizations of totally projective p-groups, it seems to be a natural question to ask for the structure of the duals of Warfield groups. The aim of this paper is, both

The minimal extension of P-localization on groups

Let P be a fixed set of primes, the category of all groups and group-homomorphisms, and the full subcategory of nilpotent groups. In [9], an idempotent functor called P-localization, was defined so as to extend the ℤ-module-theoretic localization of abelian groups. There are two well-known extensions of e to , namely, Bousfield's P-localization [2], [4], denoted by EZP, and Ribenboim's P-localization [13], usually denoted by ( )P. Ribenboim's P-localization is the maximal extension among localizations extending e to in that it maximizes the number of groups in its image [7]. The localized groups obtained after applying Ribenboim's P-localization are precisely the P-local groups, that is, the groups having unique nth-root for every n whichis co-prime to P, [13]. Being maximal is equivalent to this class of P-localized groups being the saturated class of groups generated by e–equivalences, that is, group homomorphisms between nilpotent groups which become isomorphisms after applying e.

On a family of Serre classes of nilpotent groups

Let N be a P-local nilpotent group. We say that N is finitely generated (fg) as P-local group if there is a finite subset S of N such that the smallest P-local subgroup of N containing S is N itself; we could also say that S generates N as P-local group and write N=〈 S〉 p . If G is nilpotent, we say that G is finitely generated at every prime (fgp) if G p is fg as p-local group for all primes p. Such groups share with fg nilpotent groups many important properties (e.g., they are Hopfian). We show that they satisfy the axioms for a Serre class, as extended by Hilton and Roitberg to nilpotent groups.

Some remarks on almost finitely generated nilpotent groups

We identify two generalizations of the notion of a finitely generated nilpotent. Thus a nilpotent group G is fgp if Gp is fg as p-local group for each p; and G is fg-like if there exists a fg nilpotent group H such that Gp @ Hp for all p. The we have proper set-inclusions: {fg} I {fg-like} I {fgp}. We examine the extent to which fg-like nilpotent groups satisfy the axioms for a Serre class. We obtain a complete answer only in the case that [G, G] is finite. (The collection of fgp nilpotent groups is known to form a Serre class in the extended sense).

Euler characteristics and cohomology of p-local discrete groups

Euler characteristics and cohomology of p-local discrete groups

On the structure of p-local abelian groups

A uniqueness theorem in terms of numerical invariants is established for certain mixed p-local groups, called B-modules, which include both the balanced projective groups and the torsion A-groups. In general, a B-module does not contain a decomposition basis, and when it does, it reduces to a direct sum of an A-group and a balanced projective group. A special class of B-modules, the members of which are called S-modules, constitutes a natural generalization of S-groups and the following fact is proved: If H is an isotype subgroup of the p-local balanced projective group G with G H countable, then H is an S-module.

Existence theorems for Warfield groups

Warfield studied p-local groups that are summands of simply presented groups, introducing invariants that, together with Ulm invariants, determine these groups up to isomorphism. In this paper, necessary and sufficient conditions are given for the existence of a Warfield group with prescribed Ulm and Warfield invariants. It is shown that every Warfield group is the direct sum of a simply presented group and a group of countable torsion-free rank. Necessary and sufficient conditions are given for when a valuated tree can be embedded in a tree with prescribed relative Ulm invariants, and for when a valuated group in a certain class, including the simply presented valuated groups, admits a nice embedding in a countable group with prescribed relative Ulm invariants. These conditions, which are intimately connected with the existence of Warfield groups, are given in terms of new invariants for valuated groups, the derived Ulm invariants, which vanish on groups and fit into a six term exact sequence with the Ulm invariants.