Separable P-group Research Articles

Small pairs of abelian p-groups

The aim of this paper is to introduce and investigate small pairs of abelian p-groups, defined by means of small homomorphisms and by analogy with torsion theories and cotorsion pairs. Non-trivial small pairs live inside the two classes of thick p-groups and thin p-groups. Since these groups, as well as small homomorphisms, have been investigated long time ago, in the first part of the paper we survey known results on these topics, adding some new results. A relevant role in the study of small pairs is played by the classes of fully-thick and fully-thin groups and by the groups which are at the same time thick and thin. This last class of groups is large enough to guarantee the existence of 22c different small pairs. Connections with a Corner's classical realization theorem for endomorphism rings of separable p-groups and the recent topic of groups with minimal full inertia are established.

To the Question of Full Transitivity of a Cotorsion Hull

We discuss the topic of full transitivity of the cotorsion hull of a separable p-group. This topic plays a significant role in the description of the lattice of fully invariant subgroups of a group. A condition is established, the fulfillment of which makes the cotorsion hull not fully transitive.

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

For separable p-groups, the questions concerning the full transitivity and description of the lattice of fully invariant subgroups of the cotorsion hull are discussed. We consider the case in which the cotorsion hull is not fully transitive, and construct a function, different from the indicator, that makes it possible to study the lattice of fully invariant subgroups of a cotorsion group.

The Lattice of Fully Invariant Subgroups of a Cotorsion Group

In this paper, we consider the lattice of fully invariant subgroups of the cotorsion hull of a separable p-group, which is assumed to be an arbitrary direct sum of torsion-complete p-groups. It is shown that this lattice is isomorphic to the lattice of filters of a lower semilattice made up of infinite matrices.

On the full transitivity and fully invariant subgroups of cotorsion hulls of separable p-groups

On the full transitivity and fully invariant subgroups of cotorsion hulls of separable p-groups

On the full transitivity and fully invariant subgroups of cotorsion hulls of separable p-groups

The paper deals with questions connected with the description of a lattice of fully invariant subgroups of cotorsion hulls of separable p-groups. In particular, attention is given to the full transitivity of a group, which plays an essential role in describing this lattice.

On the Full Transitivity of a Cotorsion Hull

Abstract A cotorsion hull of the separable 𝑝-group 𝑇 is considered when 𝑇 is a direct sum of torsion-complete groups. It is proved that in the considered case its cotorsion hull is fully transitive if and only if 𝑇 is a direct sum of cyclic groups or is a torsion-complete group.

Universal abelian groups

We examine the existence of universal elements in classes of infinite abelian groups. The main method is using group invariants which are defined relative to club guessing sequences. We prove, for example:Theorem:For n≧2, there is a purely universal separable p-group in ℵ n if, and only if, $$2^{\aleph _0 } \leqslant \aleph _n $$ .

Determinimg abelianp-groups from theirn-Socles

A necessary condition is given for a very wide separable p-group to be derermined by its n-socle (the set of elements of order at most p n). A group G is very wide if it has a direct summand of final rank ‖G‖ which itself is a direct sum of cyclic groups. In the case, n=1 (as Shelat has shown)this condition is sufficient but for n>1 examples are given to show that this condition is not sufficient.

The existence of certain pure dense subgroups of abelian p-groups is not decidable in ZFC

Let κ be a regular uncountable cardinal and n a positive integer. Let G be a separable p-group, and let H be a proper pure dense subgroup of G with a κ-filtration H =∪ αϵκH α, such that {αϵκ: p ω + n( H H α )≠ 0 } is stationary in κ. Assuming V = L, there are 2 κ pairwise nonisomorphic pure subgroups of G each having the same p n -socle as H. The following two questions are not decidable in ZFC. Let H be a pure dense subgroup of a separable p-group G, and let n be a positive integer. Does there exist a pure subgroup K of G such that K[ p n ] = H[ p n ] and K is not isomorphic to H? If A is an abelian p-group, are all high subgroups of A isomorphic?

The number of separable p-groups of cardinality r1 Having isometric socles

The number of separable _p-groups of cardinality r₁ Having isometric socles

The injective hull of a separable p-group as an E-module

The injective hull of a separable p-group as an E-module