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On a Theorem of Cutler

Published online by Cambridge University Press:  20 November 2018

Charles K. Megibben
Charles K. Megibben*
Affiliation:
Vanderbilt University Nashville, Tennessee
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In [1] Cutler proved the following theorem.

Theorem. If G and K are abelian groups such that nG ≅ nK for some positive integer n, then there are abelian groups U and V such that U ⊕ G ≅ V ⊕ K and nU = 0 = nV.

Cutler's proof is long and fairly involved. Walker [3] obtains the theorem rather elegantly as a corollary of his results on n-extensions. We give here a proof that is extremely simple both in conception and execution. Our proof relies on the notion of p-basic subgroups introduced by Fuchs in [2]. Therefore we shall first recall certain pertinent facts from [2].

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 12 , Issue 2 , April 1969 , pp. 225 - 227
Copyright
Copyright © Canadian Mathematical Society 1969

References

1. Cutler, D.O., Quasi-isomorphism for infinite abelian p-groups. Pacific J. Math. 16 (1966) 25–45.Google Scholar
2. Fuchs, L., Notes on abelian groups II. Acta. Math. Acad. Sci. Hungar. 11 (1960) 117–125.Google Scholar
3. Walker, E.A., On n-extensions of abelian groups. Annales Univ. Sci. Budapest 8 (1965) 71–74.Google Scholar