By Robert B. Warfield Jr. (auth.), David M. Arnold, Roger H. Hunter, Elbert A. Walker (eds.)
Read or Download Abelian Group Theory: Proceedings of the 2nd New Mexico State University Conference, Held at Las Cruces, New Mexico, December 9–12, 1976 PDF
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Additional resources for Abelian Group Theory: Proceedings of the 2nd New Mexico State University Conference, Held at Las Cruces, New Mexico, December 9–12, 1976
Sem. Abelian T-~7"2. - of elongations 289-304. of Abelian theory of Abelian groups II: for modules and the stable range 38 53. B. Wick, Classification theorems for infinite Abelian groups, Dissertation, Univ. of Washington, 1972. DECOMPOSITION BASES AND ULM'S THEOREM Robert O. Stanton i. Introduction. In 1933, Ulm  classified countable reduced primary groups via cardinal invariants. Kaplansky and Mackey  gave a new proof of Ulm's theorem, which served as a basis for future generalizations.
For primes q # p, the proof is similar, except that it may happen that the analogues to both (i) and (ii) fail to hold. Hq(X I) = Hq(X2). In this case Hq[Yl) = Hq(Y2) = Also the q-factor of m I must equal the q-factor of m 2. thus assume (ml,q) = (m2,q) = (ml,m2) = I. We can We wish to have Hq(Z) = Hq(Yl). not, then q must divide both (mla I + m2bl) and (mla 2 + m262). If This implies q divides a2b I - alb 2 = I, a contradiction. 4. The Classification Theorem. We will prove a structure theorem for a class of mixed groups containing the totally projective p-groups and the completely decomposable torsion-free groups.
Then the following three conditions are equivalent. (a) S is basis-equivalent to S'. (b) For all c, p and e, ST(c,p,e,S) (c) (i) There is a bijection ¢: = ST(c,p,e,S'). S + S' such that M and #(M) are compatible for all M e S. (ii) For each prime p, there is a bijection ~p: S + S' such that ~p(M) is compatible with M and (~p(M))p is equivalent to Mp. Moreover, ~p(M) = ¢(M) for all but finitely many primes p. Proof. Corollary 3 provides the proof of (a) implies (b). implies (c) is based on the following set theoretical argument.
Abelian Group Theory: Proceedings of the 2nd New Mexico State University Conference, Held at Las Cruces, New Mexico, December 9–12, 1976 by Robert B. Warfield Jr. (auth.), David M. Arnold, Roger H. Hunter, Elbert A. Walker (eds.)