Abelian Groups and Modules: Proceedings of the Udine by R. Göbel, C. Metelli, A. Orsatti, L. Salce

By R. Göbel, C. Metelli, A. Orsatti, L. Salce

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2. 1)). Asolutions, (i), (iv) exclude less than So we can carry on the def1nition, and then let holds. By 'II\• 1 A of (iii) requires that we have ~~'~w distinct ordinals satisfying (i), (ii), (iv). : for ~((f,N))=sup(orco(Rang(f))). every branch 1 of Rang( f), 51 A Combinatorial Theorem sup(orco(~) =]((f,N)). l+ l (cS) If ' 7 (f,N)E W. 1 :u {"l~k: ' i ~ -(f' ,N')e w'f, k(G)}SN, then (f' ,N') 1 This is straightforward; finish well-ordering the <~. 13 By (£), w'f = U:i. wll • I. t): Gl(oL*} that for ~

25 Subgroups of Bounded Abelian Groups PROOF. 5), map from to Thus we can map x COROLLARY 2. 8. direct sum of map PROOF. C. If vx a a, C such that C(a) C of order A to a direct sum of copies of As every subroup of a p-nice, the subgroup 3. direct FC(A) cyclics is 0. As ~ A(a) 0. ~ p-group, and C a A Then there is a such that = Ker FC(A) is nice in c. Every [HRWl; A. 3 says can be mapped D 2 p -bounded finite valuated Theorem 4). We extend this 2 result to p -bounded valuated groups with finitely many values.

1. Context: (1) ).. ~ thus enables us to give a uniform proof. X, ). , but also with ). ~ and cf ). X! Jf0 , for each n. n! = lC <). <>. 1. 3. e. M and a function from "'to There are a regular cardinal /"• in change T = U lT ~ = ""<'-' ~<"" demand 1. 2. Definition: E a GJ , and for some { ,g gf the class ~~).

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