Infinite abelian groups, whitehead problem and some constructions

@article{Shelah1974InfiniteAG,
  title={Infinite abelian groups, whitehead problem and some constructions},
  author={Saharon Shelah},
  journal={Israel Journal of Mathematics},
  year={1974},
  volume={18},
  pages={243-256}
}
  • S. Shelah
  • Published 1974
  • Mathematics
  • Israel Journal of Mathematics
We solve here some problems from Fuchs’ book. We show that the answer to Whitehead’s problem (for groups of power ℵ1) is independent from the usual axioms of set theory. We prove the existence of large rigid systems for groups of power λ, with no restriction on λ. We also prove that there are many non-isomorphic reduced separablep-groups. 
Finding bases of uncountable free abelian groups is usually difficult
We investigate effective properties of uncountable free abelian groups. We show that identifying free abelian groups and constructing bases for such groups is often computationally hard, depending onExpand
Whitehead groups may not be free even assuming ch, II
AbstractWe prove some theorems on uncountable abelian groups, and consistency results promised in the first part, and also that a variant of $$ \diamondsuit _{\omega _1 } $$ called ♣ (club), isExpand
A combinatorial principle equivalent to the existence of non-free Whitehead groups
As a consequence of identifying the principle described in the title, we prove that for any uncountable cardinal lambda, if there is a lambda-free Whitehead group of cardinality lambda which is notExpand
Methods of Set Theory and the Abundance of Separable Abelian p-Groups
The aim of this paper is to discuss the significance of a certain set-theoretic invariant г and the relation of quotient-equivalence for separable abelian p-groups of cardinality ωl. By means ofExpand
On the structure of Extp(G, Z)
Abstract We will prove a theorem on the cardinality of inverse limits of systems of groups. The following is an instance of the theorem: Theorem . Let λ be a strong limit cardinal of cofinality ℵ 0 .Expand
Indecomposable explicit abelian group
For every lambda we give an explicit construction of an Abelian group with no non-trivial automorphisms. In particular the group absolutely has no non-trivial automorphisms, hence is absolutelyExpand
A compactness theorem for singular cardinals, free algebras, Whitehead problem and tranversals
We prove, in an axiomatic way, a compactness theorem for singular cardinals. We apply it to prove that, for singular λ, every λ-free algebra is free; and similar compactness results for transversalsExpand
EVERY COTORSION-FREE RING IS AN ENDOMORPHISM RING
Some years ago A. L. S. Corner proved that every countable and cotorsion-free ring can be realized as the endomorphism ring of some torsion-free abelian group. This result has many interestingExpand
Absolutely rigid systems and absolutely indecomposable groups
We give a new proof that there are arbitrarily large indecomposable abelian groups; moreover, the groups constructed are absolutely indecomposable, that is, they remain indecomposable in any genericExpand
On uncountable abelian groups
We continue the investigation from [10], [11], [12] on uncountable abelian groups. This paper tends more to group theory and was motivated by Nunke’s statement (in [9]) that Whitehead problem,Expand
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 13 REFERENCES
Intersection Theorems for Systems of Sets
A version of Dirichlet's box argument asserts that given a positive integer a and any a2 +1 objects x0 , x1 , . . ., xa 2, there are always a+1 distinct indices v (0 < v < a 2) such that theExpand
Intersection theorems for systems of sets (ii)
In this paper we present the complete solution of the problem which was considered in [1], with the exception of the case in which both the given cardinal numbers are finite. The results of [1] willExpand
The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.
  • K. Gödel
  • Mathematics, Medicine
  • Proceedings of the National Academy of Sciences of the United States of America
  • 1938
TLDR
Kurt Godel, mathematician and logician, was one of the most influential thinkers of the twentieth century and ranked higher than fellow scientists Edwin Hubble, Enrico Fermi, John Maynard Keynes, James Watson, Francis Crick, and Jonas Salk. Expand
Abstract Harmonic Analysis
The first € price and the £ and $ price are net prices, subject to local VAT. Prices indicated with * include VAT for books; the €(D) includes 7% for Germany, the €(A) includes 10% for Austria.Expand
The fine structure of the constructible hierarchy
Infinite Abelian groups
Internal cohen extensions
...
1
2
...