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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Extra resources for Abelian Groups
14. Further criteria for the existence of a basis Here we prove three theorems concerning the existence of a basis: the first is applicable to p-groups, the second to countable torsion free groups, while the third one applies to arbitrary groups. 1. ) A p-group G containing no elements Φ 0 of infinite height is a direct sum of cyclic groups if and only if G contains a principal system. Here we mean by a principal system a maximal independent set L = [a>]xeA no element of which can be replaced by an element g(d G) of a greater height without violating independence.
For the proof of the sufficiency assume that the Gn satisfy the hypoth eses, and form the socle Pn of G„. Select in Anp f c i G a maximal inde pendent set of elements, and expand it in turn with elements of Pifip A i I G, ... . , PifipG, Pi to an independent set Si whicn is in each step maximal. Next proceed to P2 and extend Si in P2np**G, then in P o f i p ^ G , . . , PidpG, Po, so that the independent set obtained after each step is maximal. Then repeat this process with the set S2 thus constructed etc.
13. ) Every group G has a generating system [g)]keA such that each g ζ G admits a unique expression g^^g^ -\ \~ nkgk where the coefficients n of gk are subject, for a fixed Λ, to one of the conditions: 1) n is an arbitrary integer, 2) n assumes only the values 0 , 1 , . . , / π λ — 1 (ηιλ fixed). [Hint: as in Ex. ] 14. (a) Each countably generated p-group is a homomorphic image of uFp(l). (b) Give explicitly a subgroup N of $>(1) such that §P(\)/Ncx(2(p<*>). • 15. Define the concept of a free >H-module with an arbitrary associative ring % as operator domain and extend the results of § 9 .