G.C.H. implies existence of many rigid almost free abelian groups

The First Almost Free Whitehead Group Sh914

Assume G.C.H. and κ is the first uncountable cardinal such that there is a non-free κ-free abelian Whitehead group of cardinality κ. We prove that if all κ-free Abelian group of cardinality κ are Whitehead then κ is necessarily an inaccessible cardinal.

On the Existence of Rigid א 1 – Free Abelian Groups of Cardinality

Almost free groups and Ehrenfeucht-Fräıssé games for successors of singular cardinals

We strengthen non-structure theorems for almost free Abelian groups by studying long Ehrenfeucht-Fräıssé games between a fixed group of cardinality λ and a free Abelian group. A group is called ǫ-game-free if the isomorphism player has a winning strategy in the game (of the described form) of length ǫ ∈ λ. We prove for a large set of successor cardinals λ = μ the existence of nonfree (μ · ω1)-g...

Localizations of Groups

A group homomorphism η : A → H is called a localization of A if every homomorphism φ : A → H can be ‘extended uniquely’ to a homomorphism Φ : H → H in the sense that Φη = φ. This categorical concepts, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the intr...

Constructing Simple Groups for Localizations

A group homomorphism η : A → H is called a localization of A if every homomorphism φ : A→ H can be ‘extended uniquely’ to a homomorphism Φ : H → H in the sense that Φη = φ. This categorical concept, obviously not depending on the notion of groups, extends classical localizations as known for rings and modules. Moreover this setting has interesting applications in homotopy theory, see the introd...