By laszlo fuchs

ISBN-10: 0080092063

ISBN-13: 9780080092065

Abelian teams offers with the idea of abelian or commutative teams, with distinctive emphasis on effects relating constitution difficulties. greater than 500 workouts of various levels of hassle, with and with out tricks, are incorporated. a few of the routines remove darkness from the theorems brought up within the textual content by means of supplying substitute advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts off with an outline of the fundamental proof on team thought similar to issue staff or homomorphism. The dialogue then turns to direct sums of cyclic teams, divisible teams, and direct summands and natural subgroups, in addition to Kulikovs uncomplicated subgroups. next chapters concentrate on the constitution conception 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 jewelry; the Schreier extension concept with a dialogue of the crowd of extensions and the constitution of the tensor product. moreover, the booklet examines the speculation of the additive workforce of earrings and the multiplicative team of fields, besides Baers conception 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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Then H is the union of the ascending chain Ηι^Ηο^... with Hn = HnGn and the heights of the elements of Hn, taken in //, do not exceed kH. Application of Theorem 11. 1 shows that H is a direct sum of cyclic groups. 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.

Then r ( / / ) ^ / · ^ ) + + ---+/W (b) H = Hl-\ 1- // n implies equality in the last inequality. (c) Prove that (b) holds for the ranks r0(H) and rp(H), but not for the reduced rank. 33. No subgroup of G can be decomposed into the direct sum of two of its proper subgroups if and only if r(G) = l. 24 35. The socle S(G) of G consists of all elements whose orders are square free integers. Show that S(G)=--^>\ G[p], the sum extended over all primes p. 36. , Hn are subgroups of G such that each G/Hi is of finite rank, then G/H with / / = / / 1 η · · · Π / / η is again of finite rank.

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. (b) This fails to hold if H is of infinite rank. 7. ) Which groups G have the following property: if Z/^-G, then G is imbeddable in / / ?

### Abelian Groups by laszlo fuchs

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