Abelian Groups by laszlo fuchs

By laszlo fuchs

Abelian teams offers with the speculation of abelian or commutative teams, with targeted emphasis on effects referring to constitution difficulties. greater than 500 workouts of various levels of hassle, with and with no tricks, are incorporated. a few of the workouts remove darkness from the theorems brought up within the textual content via supplying replacement advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts off with an summary of the elemental evidence on workforce idea similar 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 uncomplicated 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 speculation also are thought of, in addition to different subject matters reminiscent of 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 booklet examines the idea of the additive staff of earrings and the multiplicative team of fields, in addition to Baers idea of the lattice of subgroups.

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

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Let G be a subgroup of the direct sum A + B. (a) Then there exist subgroups Ax, A2 of A and subgroups Bt fB2oiB such that A1^A2,B1^B2 and Al-\-Bl is the minimal direct sum containing G and A2 + B2 is the maximal direct sum contained in G with components in A and B, respectively. 2^BllB2 and (Ax + Bx)/G o*t G/(A2 + B2). 18. ) (a) Let G = A + B = C*+D. Then {A, C} - - Λ + B = C + Π where B' is the set of all /^-components of C and U is the set of all D-components of A. ] (b) G = A + B==C + D implies A;{A[\C)^D' and C/(A n C)c*B'.

25. (a) A cyclic sJl-module {a}^ with 0(a) = 0 is sJt-isomorphic to the additive group 91+ of fit. If Ο ( ο ) φ 0 , then {a}s)t is 9t-isomorphic to the ^-module

Extend this statement to the p-components of T. 13. (a) The multiplicative group of all non-zero complex'numbers is the direct prod­ uct of the multiplicative group of all positive real numbers and that of all complex numbers with absolute value 1. (b) The additive group A of all complex numbers is the direct sum of two groups, each isomorphic to the group of all reals. In how many ways can A be so decomposed? 14. Let G = A + B = A + C; then the ^-components of an element g£ G are not necessarily the same in the two direct decompositions.

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