A Countability Condition for Primary Groups Presented by Relations of Length Two by Paul Hill
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چکیده
A subgroup A of the ^-primary group G is nice if p(G/A) = {pG, A }/A for all ordinals a. We consider the following countability condition: there exists a collection G of nice subgroups of G such that (0) oee. (1) 6 is closed with respect to group-theoretic union in G. (2) If A^e and if H is a subgroup of G such that {Af H}/A is countable, there exists i ? £ G such that 5 2 {-4, H\ and such that B/A is countable. The author [ l ] has referred to this condition as the third axiom of countability and has demonstrated that this is the countability condition—not the first axiom (countability) nor the second axiom (decomposition into a direct sum of countable groups)—which is truly relevant for the proof of Ulm's theorem. In this note, we outline a short proof of
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