laszlo fuchs's Abelian Groups PDF

By laszlo fuchs

ISBN-10: 0080092063

ISBN-13: 9780080092065

Abelian teams offers with the speculation of abelian or commutative teams, with distinct emphasis on effects pertaining to constitution difficulties. greater than 500 workouts of various levels of hassle, with and with no tricks, are integrated. many of the workouts remove darkness from the theorems pointed out within the textual content through delivering substitute advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts with an summary of the elemental evidence on crew thought comparable 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 specialize in the constitution concept of the 3 major sessions of abelian teams: the first teams, the torsion-free teams, and the combined teams. functions of the speculation also are thought of, besides 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 speculation of the additive team of jewelry and the multiplicative staff of fields, besides Baers conception of the lattice of subgroups.

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

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Example text

2 is true for torsion groups. Now if G is an arbitrary direct sum of cyclic groups, T is its maximal torsion subgroup, then / / n T is the maximal torsion subgroup of the sub­ group H of G and ///(//n 7 ) ^ { / / , T}/T is isomorphic to some subgroup of the free group GIT. Hence ///(//n T) is free by Theorem 12. 2, H is the direct sum of Hn T and a free group. In the pre­ ceding paragraph we have shown that Hn T is a direct sum of cyclic groups. This completes the proof. 3. ) Any two direct decompositions of a group which is a direct sum of cyclic groups have isomorphic refinements.

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\. Then we have /i|/; ( / = 1 , . . , n). 9αΗ is a basis of sF(n), and the extremal property assumed for ai9.. 9an implies r2= ··· =ru = 0. n ζ {α2} + · · · + {an}. It results: G = {61} + {a2} + ·· · • · · + {flu}, V= {vx} + V" where n = h öi and V" c {α2} Η |- {a,,}. A simple induction leads to the conclusion that for appropriate decompositions G = {61}+ .

Then the correspondence flxn-*gV induces a homomorphism of ^ ( m ) onto G. ) § 10. Finite and finitely generated groups The next theorem, often called the Basis Theorem, has been the first really substantial structural result in the theory of groups. It has many appli­ cations in number theory and other parts of mathematics. Its extension to larger classes of groups has been one of the major aims in this theory. 1. ) A finite group is the direct sum of a finite number of cyclic groups of prime power order.

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Abelian Groups by laszlo fuchs

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