Download PDF by laszlo fuchs: Abelian Groups

By laszlo fuchs

ISBN-10: 0080092063

ISBN-13: 9780080092065

Abelian teams bargains with the speculation of abelian or commutative teams, with precise emphasis on effects touching on constitution difficulties. greater than 500 routines of various levels of trouble, with and with out tricks, are integrated. a number of the workouts remove darkness from the theorems stated within the textual content by means of offering substitute advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts with an summary of the elemental evidence on crew idea similar to issue team or homomorphism. The dialogue then turns to direct sums of cyclic teams, divisible teams, and direct summands and natural subgroups, in addition to Kulikovs easy subgroups. next chapters concentrate on the constitution idea of the 3 major periods of abelian teams: the first teams, the torsion-free teams, and the combined teams. functions of the idea also are thought of, besides different themes equivalent to homomorphism teams and endomorphism jewelry; the Schreier extension thought with a dialogue of the gang of extensions and the constitution of the tensor product. furthermore, the booklet examines the idea of the additive team of earrings and the multiplicative crew of fields, in addition to Baers concept of the lattice of subgroups.

This ebook is meant for younger learn staff and scholars who intend to familiarize themselves with abelian teams.

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If αλ is of infinite order, then ft is of a greater order than a} if rä = stf ; 4=0 holds with M>|*|. L. Fuchs: Abelian Groups 50 DIRECT SUM OF CYCLIC GROUPS [Chap. II pl(t^0) be the greatest power dividing every nr, put ni=ptnl and assume e. g. that (n\, p) = 1. Then b = n[ αλχ H \-n'k axk satisfies 0(b) ^pl< 0(αλι) and, since (n[,p)=\t if we replace αλι by b in By we shall again get a generating system for G. We have thus derived a contradiction to

Let ^=ΖΣ* Σ^^Ρ^ with a fixed k. Prove that G is a (discrete) direct sum of in n groups Q(pk) and determine the cardinality of the set of the components. 53. ) Let G b e a bounded p-group and p r the least upper bound for for all a£G if and only if G is a direct sum of the orders of its elements. a£[pr/0(a)]G cyclic groups of order pr. (b) If the elements of G[p] have the same height r—1, then G is a direct sum of cyclic groups of order p r . 54. (a) Determine those groups in which every maximal independent set is a basis.

4 to M-modules over principal ideal rings M. • 44. Show that not every W-submodule of a finitely generated W-module is finitely generated, if M is arbitrary. ] 45. )ir> Prove that if G = A + H = B+J where A and B are isomorphic finitely generated groups, then H o*J. [Hint: assume H*J\ if A = g (^ ), then H/(Hf]J) = = (2 (<~) and H = (2(DO) + (Hf]J); if A = {a} = Q (p*) and p*-i a $ / , then G = A +J, and if ρ*-ι α £ / , £ = { £ } , ρ * - ι δ ζ / / , then G = {a + b} + H={a + b\ + / . ] 46. Prove the isomorphism of the direct decompositions of finitely generated groups by making use of Ex.

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