# 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} }

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.

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#### References

SHOWING 1-10 OF 13 REFERENCES

Intersection Theorems for Systems of Sets

- Mathematics
- 1960

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 the… Expand

Intersection theorems for systems of sets (ii)

- Mathematics
- 1969

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] will… Expand

The Consistency of the Axiom of Choice and of the Generalized Continuum-Hypothesis.

- Mathematics, Medicine
- Proceedings of the National Academy of Sciences of the United States of America
- 1938

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

- Mathematics
- 1963

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