New PDF release: Abelian Groups

By laszlo fuchs

ISBN-10: 0080092063

ISBN-13: 9780080092065

Abelian teams bargains with the idea of abelian or commutative teams, with detailed emphasis on effects pertaining to constitution difficulties. greater than 500 workouts of various levels of hassle, with and with out tricks, are incorporated. the various workouts remove darkness from the theorems mentioned within the textual content by way of delivering replacement advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts off with an outline of the elemental proof on team thought resembling 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 simple subgroups. next chapters specialise in the constitution thought 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, in addition to different issues akin to homomorphism teams and endomorphism earrings; the Schreier extension idea with a dialogue of the gang of extensions and the constitution of the tensor product. additionally, the ebook examines the idea of the additive crew of earrings and the multiplicative workforce of fields, in addition to Baers thought of the lattice of subgroups.

This e-book is meant for younger study staff and scholars who intend to familiarize themselves with abelian teams.

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0(ak) assumes its maximum (minimum). 20. :p* r is equal to P(k^.. P(kt) where P(k) is the number of partitions of k into positive integers. (b) When is this number equal to 1? 21. Let S be the set consisting of all elements in a finite group G which are of maximal order. Then {5} = G. 22. (a) If m divides the order of G, then G has subgroups as well as factor groups of order m. (b) A group H is isomorphic to some subgroup of a finite group G if and only if it is isomorphic to some factor group of G.

2. A finitely generated group G is a homomorphic image of §y(n) if and only if the reduced rank of G does not exceed n. 3. Let H and J be subgroups of a free group. Show that r(HdJ) + r ({//,/}) = = r(H) -f r(J). ) 4. Theorem 9. 2 remains no longer valid if G/H is a direct sum of finite cyclic groups. 5. 'Then H(\J is a direct summand of H. ] (b) The meet of a finite number of direct summands of a free group is again a direct summand. 6. (a) A subgroup H of finite rank of a free group F is a direct summand of F if and only if FjH is torsion free.

49. G is elementary if and only if it is a torsion group and its φ-subgroup vanishes. 50. Let $ be a field of characteristic p (Φ 0). (a) The additive group of £ is an elementary p-group. (b) Prove the same for the polynomial ring Ä[x]. (c) Extend (b) to an arbitrary ring Ή by proving that the additive group of ))i[x] is a direct sum of a countable set of groups isomorphic to the additive group of N. 51. For any G, G [n] is a direct sum of cyclic groups. 52. 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.

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Abelian Groups by laszlo fuchs

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