/pdf/modules-over-dedekind-rings-and-valuation-rings-5f2hjvdkub.pdf

Abelian groups without elements of finite order

Modules over Dedekind rings and valuation rings

Almost free modules

Die Folgen ganzer Zahlen {an} (vom Typus ω) bilden auf Grund der Addition {an} + {bn} = {an + bn} eine abelsche Gruppe F. Diese Gruppe F und gewisse ihrer Untergruppen sollen im folgenden untersucht werden. Als erstes werden wir zeigen, dass jede abzahlbare Untergruppe von F eine freie abelsche Gruppe ist (Satz I; eine abelsche Gruppe heisst eine freie abelsche Gruppe, wenn sie eine solche Teilmenge B — Basis genannt — besitzt, dass jedes Gruppenelement J darstellbar ist in der Form \(\mathcal{J} = \sum\limits_{i = 1}^m {{n_i}{b_i}}\), ni ganz, bi ∈ B und wenn aus Σ nibi = 0, bi∈ B, bi ≠ b, fur i ≠ j folgt, dass ni = 0; da alle betrachteten Gruppen abelsch sind, werden wir statt «freie abelsche Gruppe» auch kurz «freie Gruppe» sagen.) Die Gruppe F selbst dagegen ist nicht frei. Wir werden namlich zeigen, dass F eine nicht freie Untergruppe der Machtigkeit ℵ1 besitzt (Satz II, der dasselbe auch fur gewisse Untergruppen von F aussagt); die F selbst betreffende Behauptung ist darin enthalten, da jede Untergruppe einer freien Gruppe frei ist (vgl. z. B. S. Lefschetz [1], S. 50).

Additive Gruppen von Folgen Ganzer Zahlen

Chapter Chapter Chapter Chapter I . Def in i t ions a n d genera l i t ies . . . . . . . . . . . . . . . . . . . 222 II . Local ly n i l p o t e n t g roups Z A g r o u p s a n d n i l p o t e n t g roups . . . . . . 229 I I I . C o n s t r u c t i o n of U~-groups , E~groups a n d D~-groups . . . . . . . . 237 VI. E x t e n s i o n s of U~groups , E~-g roups a n d D~-groups . . . . . . . . . 246

/pdf/some-aspects-of-groups-with-unique-roots-1zmw74ow92.pdf

Some aspects of groups with unique roots

1. Introduction. Elsewhere(2) we consider the problem of characterizing all the solutions x, in a free group on generators a,, a2, . . *, a., of a given equation w(x, a,, a2, , a.) = 1. The totality of solutions can be described as the set of all values assumed by a certain finite set of group theoretic expressions upon substituting integers for certain parameters appearing in these expressions as exponents. As a trivial example, the solutions of x-lajlxa1= 1 are all values assumed by a' as v runs through the integers. The general result, and our method of obtaining it, has led us to the study of such expressions, or "words," that contain certain parameters Pi, P2, * * *, Vd as exponents. These words may be taken in a natural way as representing elements of a group G that admits additional algebraic operations of raising an element g of G to an exponent a, where a is any element of the ring Z[P1, P2, * * *, Pd] of all polynomials in indeterminates in Pi, P2, * * *, Pv with integer coefficients. More generally, if X is any associative ring with 1, we call a group G an X-group if it is equipped with additional operations g-_g for each a in X, subject to the following axioms: g' = g, g (a+:) = gag 3, g (a0) = (g a) a g(hg) a = (gh) ag. We hasten to note that if n is an integer, the axioms imply g n = gn, so that we may omit all dots. Like the power maps, g y+gn, these operations are not required to define endomorphisms: we do not require that (gh)a=gah/a. The last axiom requires that, like the power maps, these operations commute with all inner automorphisms: (g-'hg) a = g-lhag. The appropriateness of these axioms to our purpose is demonstrated by the fact, proved below, that for X=Z[Pl, V2, P, Pd], a word w in the letters a,, a2, * * *, a. containing parameters Pi, P2, * d, assumes the value 1 in the free group on a1, a2, * * *, ar under all substitutions for the parameters if and only if it reduces to the "empty word" 1 by virtue of the given axioms. The main result of this paper is the solution of an extended word problem for free X-groups, X=Z[Pl, P2, * * *, Pd]. An effective process is exhibited whereby, given a word w, as above, an element a in X is determined such that, under any substitution of integers for the vi, co assumes the value 1 if and only if a assumes the value 0.

background

/pdf/groups-with-parametric-exponents-37fgmvp68c.pdf

Abelian groups without elements of finite order

Citations

Modules over Dedekind rings and valuation rings

Almost free modules

Additive Gruppen von Folgen Ganzer Zahlen

Some aspects of groups with unique roots

Groups with parametric exponents

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