Vol. 107, No. 1, 1983

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Crawley’s problem on the unique ω-elongation of p-groups is undecidable

Charles Kimbrough Megibben, III

Vol. 107 (1983), No. 1, 205–212

Let G be an abelian p-group with pωG = 0. Crawley has raised the following question: If all groups A with pωA cyclic of order p and A∕pωAG are mutually isomorphic, is G necessarily a direct sum of cyclic groups? We show this question to be independent of the axioms of set theory. Specifically, we prove that MA+¬CH implies a negative answer for some G of cardinality 1; whereas, if V = L is assumed, then every such G of cardinality 1 must be a direct sum of cyclic groups.

Mathematical Subject Classification 2000
Primary: 20K10
Secondary: 03D35, 03E35, 20A15
Received: 26 October 1981
Published: 1 July 1983
Charles Kimbrough Megibben, III