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

Charles Kimbrough Megibben, III

Vol. 107 (1983), No. 1, 205–212

DOI: 10.2140/pjm.1983.107.205

Abstract

Let G be an abelian p-group with p^ωG = 0. Crawley has raised the following question: If all groups A with p^ωA cyclic of order p and A∕p^ωA≅G 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 ℵ₁; whereas, if V = L is assumed, then every such G of cardinality ℵ₁ must be a direct sum of cyclic groups.

Mathematical Subject Classification 2000

Primary: 20K10

Secondary: 03D35, 03E35, 20A15

Milestones

Received: 26 October 1981

Published: 1 July 1983