# Abelian Groups by L. Fuchs,J. P. Kahane,A. P. Robertson, et PDF By L. Fuchs,J. P. Kahane,A. P. Robertson, et al.Elsevier|Elsevier Science||Pergamon Flexible LearningAdult NonfictionMathematicsLanguage(s): EnglishOn sale date: 16.12.2014Street date: 15.07.2014Preview

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E. l=pj for s o m e / £ 3 . Now p(x—ja) = h£H, but x—ja\$Hy so that {H,x—ja} contains because of the maximality of //anon-zero element ra of {a}. Thus ra = tt + s(x—ja) (Α'ζ//), sx £ H+ {a}, where we must have (s,p)=\, since p(x— ja) ζ H and {a}nH = 0. But sx,px£G* and (s,p)=\ imply x£G\ a contradiction. Thus G = {a} + H. 40 DIRECT SUM OF CYCLIC GROUPS Now ment a of applied to Next prove (Chap. 1 is obvious. We choose in G an ele­ a maximal order pk and write G = {a} + H. The same process H which is of a smaller order than G, etc.

The p-rank rp(G) of G may be defined, analogously, by using elements whose orders are powers of p rather than elements of infinite order. THEOREM (2) 8. 2. The ranks r(G), r0(G) and rp(G) are invariants of G and r(G) = ro(G)+ Σ rp(G). p=2,3,5,... ) are independent sets maximal with respect to the property of containing elements of infinite resp. prime power order. (2) implies that it suffices to verify the statement of Theorem 8. 2 only for the ranks r0(G) and rv{G). 1. In order to establish the invariance of Ab(G), we first reduce the proof to torsion free groups by showing that r0(G) = r(G/7) where T is the maximal torsion subgroup of G.

Assume that we 00 have A = Σ An where A„ is a direct sum of cyclic groups of the same n—l ω order p \ The socles Pn = Σ A· [p] form with increasing n a descending chain. Clearly, Pn consists of all those elements of P=A[p] which are of height ^n— 1. ) is p. 9 We conclude this section with the following unicity assertion. 4. Any two decompositions of a group G into direct sums of cyclic groups (of order infinity and/or prime power) are isomorphic. Denote by Pn the subgroup of P=G[p] which consists of elements of height ^n—1.