Abelian Groups - download pdf or read online

By laszlo fuchs

ISBN-10: 0080092063

ISBN-13: 9780080092065

Abelian teams bargains with the idea of abelian or commutative teams, with specific emphasis on effects referring to constitution difficulties. greater than 500 workouts of various levels of trouble, with and with no tricks, are integrated. many of the routines remove darkness from the theorems brought up within the textual content by way of offering replacement advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts off with an summary of the fundamental evidence on workforce idea corresponding 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 simple subgroups. next chapters specialise in the constitution idea of the 3 major sessions of abelian teams: the first teams, the torsion-free teams, and the combined teams. purposes of the idea also are thought of, in addition to different themes equivalent 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. furthermore, the publication examines the speculation of the additive workforce of jewelry and the multiplicative workforce of fields, besides Baers concept of the lattice of subgroups.

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

Show description

By laszlo fuchs

ISBN-10: 0080092063

ISBN-13: 9780080092065

Abelian teams bargains with the idea of abelian or commutative teams, with specific emphasis on effects referring to constitution difficulties. greater than 500 workouts of various levels of trouble, with and with no tricks, are integrated. many of the routines remove darkness from the theorems brought up within the textual content by way of offering replacement advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts off with an summary of the fundamental evidence on workforce idea corresponding 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 simple subgroups. next chapters specialise in the constitution idea of the 3 major sessions of abelian teams: the first teams, the torsion-free teams, and the combined teams. purposes of the idea also are thought of, in addition to different themes equivalent 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. furthermore, the publication examines the speculation of the additive workforce of jewelry and the multiplicative workforce of fields, besides Baers concept of the lattice of subgroups.

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

Show description

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Extra resources for Abelian Groups

Sample text

The converse statement is evident. ) are 00 f °°> Pt p2,q? Two finitely generated groups are isomorphic if and only if their invariants are the same. § 1 1 . Direct sums of cyclic p-groups If we omit the hypothesis that G be finitely generated, then it is no longer true that G is a direct sum of cyclic groups, not even for p-groups. Therefore, it is of importance to be in a possession of tools for deciding whether or not a given group may be written as a direct sum of cyclic groups. In this section we shall deal with this question in the case of p-groups (and hence torsion groups).

J be a free set of generators. If ax->gx is a single valued mapping of S into O, then it induces the correspondence //: / i i ^ H i "f- /ΪΑ-gxA. —yn1gxl H and preserves addition. The converse follows by letting the elements ax cor­ respond to the elements χλ of a free set of generators [χχ] of a free group. In particular, mapping 5 onto a generating system of G, we obtain that every group with m generators is a homomorphic image of $F (m). Let G^&(m)/N where N is the group of all linear combinations Ήι^λ,Η h^kCixk mapped upon the zero element of G.

Now mbj = 0 and it results that G = {al9... . , 0,-1, aj9 . . , bk) with O(bj) ^ /n ^ / n , < 0(aj) which contradicts the definition of the a*. For a second proof4 we show LEMMA 10. 6. Let V be a non-zero subgroup of a free abelian group §(n) of finite rank n. i\ki (i = 2 , . . , / * ) . ,an of &(n) with the following extremal property: V has an element vi = Λtfi+ · · · + L a* with a positive coefficient Λ which is as small as possible. That is, for another basis of &{n)9 or for another arrange­ ment of the aif or for other elements v of V, the leading positive coefficient cannot be less than l\.

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Abelian Groups by laszlo fuchs


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