You are here
Home > Abstract

Abelian Groups by laszlo fuchs

By laszlo fuchs

Abelian teams offers with the idea of abelian or commutative teams, with targeted emphasis on effects touching on constitution difficulties. greater than 500 routines of various levels of hassle, with and with out tricks, are integrated. a few of the workouts light up the theorems stated within the textual content via delivering substitute advancements, proofs or counterexamples of generalizations.

Comprised of sixteen chapters, this quantity starts with an summary of the elemental evidence on crew idea akin to issue crew 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 specialise in the constitution conception 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 reminiscent of homomorphism teams and endomorphism earrings; the Schreier extension thought with a dialogue of the crowd of extensions and the constitution of the tensor product. furthermore, the booklet examines the idea of the additive workforce of jewelry and the multiplicative crew of fields, besides Baers idea of the lattice of subgroups.

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

Show description

Read Online or Download Abelian Groups PDF

Best abstract books

The Selberg Trace Formula for PSL(2,R) (volume 1)

Over the last 10 years or so, mathematicians became more and more desirous about the Selberg hint formulation. those notes have been written to aid therapy this case. Their major function is to supply a complete improvement of the hint formulation for PSL(2,R). quantity one offers solely with the case of compact quotient area.

Singularities and groups in bifurcation theory.

This quantity applies pre-existing recommendations from singularity thought, specially unfolding concept and class concept, to bifurcation difficulties. this article is the 1st in a quantity series and the focal point of this e-book is singularity thought, with team idea enjoying a subordinate position. the purpose is to make singularity concept extra on hand to utilized scientists in addition to to mathematicians.

Foundations of Galois Theory (Dover Books on Mathematics)

The 1st half explores Galois thought, targeting comparable recommendations from box concept. the second one half discusses the answer of equations by means of radicals, returning to the overall conception of teams for suitable proof, interpreting equations solvable by way of radicals and their development, and concludes with the unsolvability by way of radicals of the final equation of measure n is higher than 5.

Extra info for Abelian Groups

Sample text

15· Divisibility by integers in groups The multiplication of the elements of a group by rational integers is always defined, therefore it is possible to attach a meaning to divisibility of group elements by integers. Let a be some element of a group G and n a rational integer. We shall say a is divisible by n, in sign: n\a,1 if there is an element x ( G such that nx = a, or equivalently, if a belongs to nG. We mention the following simple facts concerning divisibility. (i) The solution x of the equation nx = a is in general not unique, since together with x, exactly the elements y of the coset x+G[fl] constitute the set of all solutions y of the equation ny = a.

G is cyclic if and only if every subgroup is of the form nG. 37. ) If G is an infinite group all of whose non-zero subgroups are isomorphic to G, then G ^ (£ ( χ ) . 38. ) An infinite group G is cyclic if and only if all of its non-zero subgroups are of finite index. ] 39. If both G/H and H are finitely generated then so is G. 40. For a torsion free group to be a free group of finite rank it is necessary and sufficient that all of its torsion homomorphic images shall be finite. ) 41. A finitely generated group G may have a proper subgroup isomorphic to G, but no such a proper factor group.

I) The solution x of the equation nx = a is in general not unique, since together with x, exactly the elements y of the coset x+G[fl] constitute the set of all solutions y of the equation ny = a. (ii) If G is torsion free, then in view of (i) we see that the "quotient" nrla (if it exists) is unique. (iii) One often uses the fact that n\a if n is relatively prime to the order m of a. This is indeed true,for if s, / are integers with ms + nt=\, then x = ta satisfies nx= nta = msa + nta = a. (iv) If n\a and m\a} then also [n, m)\a.

Download PDF sample

Rated 4.10 of 5 – based on 40 votes