On Torsion-Free Abelian k-Groups (original) (raw)
Related papers
Krull Dimension and Deviation in Certain Parafree Groups
Communications in Algebra, 2010
Hanna Neumann asked whether it was possible for two non-isomorphic residually nilpotent finitely generated (fg) groups, one of them free, to share the lower central sequence. G. Baumslag answered the question in the affirmative and thus gave rise to parafree groups. A group G is termed parafree of rank n if it is residually nilpotent and shares the same lower central sequence with a free group of rank n. The deviation of a fg parafree group G of rank n is the difference µ(G) − n, where µ(G) is the minimum possible number of generators of G. Let G be fg; then Hom(G, SL(2, C)) inherits the structure of an algebraic variety, denoted by R(G), which is an invariant of fg presentations of G. If G is an n generated parafree group, then the deviation of G is 0 iff Dim(R(G)) = 3n. It is known that for n ≥ 2 there exist infinitely many parafree groups of rank n and deviation 1 with non-isomorphic representation varieties of dimension 3n. In this paper it is shown that given integers n ≥ 2, and k ≥ 1, there exists infinitely many parafree groups of rank n and deviation k with non-isomorphic representation varieties of dimension different from 3n; in particular, there exist infinitely many parafree groups G of rank n with Dim(R(G)) > q, where q ≥ 3n is an arbitrary integer. Structure of paper. New results in this paper are Theorem 1, Theorem 2, Theorem 3, Theorem 4, and Theorem 5. This paper is broken up into 3 parts: Introduction, Section one, Section two. In the introduction an outline of the mentality that guided this investigation is given, along with the proof of some preliminary results including the proof of Theorem 2, and Theorem 3. In Section One material involving sequences of primes, and groups associated with such sequences is developed. The section ends with a proof of Theorem 4. Section Two begins with a proof of Theorem 1, and ends with a proof of Theorem 5.
The balanced-projective dimension of abelian ppp-groups
Transactions of the American Mathematical Society, 1986
The balanced-projective dimension of every abelian p-group is determined by means of a structural property that generalizes the third axiom of countability. As a corollary to our general structure theorem, we show for A = Wn that every pA-high subgroup of a p-group G has balanced-projective dimension exactly n whenever G has cardinality N n but pAG *" O. Our characterization of balanced-projective dimension also leads to new classes of groups where different infinite dimensions are distinguished. O. Introduction. We consider throughout p-primary abelian groups, or equivalently, torsion modules over the integers localized at p. With possible generalizations in mind, we shall refer to them as "modules". Recall that a submodule N of M is said to be isotype if paM n N = paN for all ordinals a, and nice if < paM, N)jN = pa(MjN) for all a.
Separable abelianp-groups having certain prescribed chains
Israel Journal of Mathematics, 1990
An abelian p-group G is called p~+l-projective if p~+lExt(G,X) = 0 for all groups X. This class of groups constitutes a natural extension of the well-known class of totally projective groups whose members are precisely those groups classifiable by the Ulm-Kaplansky invariants. Fuchs asked whether p~+l-projective groups G can be characterized in terms of filtrations of G. Our Theorem 1 provides counterexamples.
Acta Mathematica Academiae Scientiarum Hungaricae
w 5. p.basic subgroups of arbitrary abelian groups KULIKOV [8] introduced the notion of basic subgroups of abelian p-groups which has proved to be one of the most important notions in the theory of p-groups of arbitrary power. Basic subgroups can be defined in any module over the ring of p-adic integers, or, more generally, over any discrete valuation ring. Here we want to give a generalization of basic subgroups to any group so that it coincides with the old concept whenever the group is primary. In the general case, to every prime p, one can define p-basic subgroups where in the definition the prime p plays a distinguished role. The p-basic subgroups are not isomorphic for different primes, but are uniquely determined (up to isomorphism) by the group and the prime p. We shall see that p-basic subgroups are useful in certain investigations. Let G be an arbitrary (abelian) group l and p an arbitrary, but fixed prime. We call a subset [x~]~ea of G, not containing 0, p-independent, if for any finite subset xl .... ,x~ a relation nlxl-[-... q-nkx1~ EprG
Balanced Subgroups of Finite Rank Completely Decomposable Abelian Groups
Transactions of The American Mathematical Society, 1987
It is proved that, if a finite rank completely decomposable group has extractable typeset of cardinality at most 5, all its balanced subgroups are also completely decomposable. Balanced Butler groups with extractable typeset of size at most 3 are almost completely decomposable and decompose into rank 1 and/or rank 3 indecomposable summands. We also construct an indecomposable balanced Butler group whose extractable typeset is of size 4 which fails to be almost completely decomposable.
An Indecomposable Balanced Subgroup of a Finite Rank Completely Decomposable Group
Journal of the London Mathematical Society, 1984
In this note only abelian groups will be considered. For general results, terminology and notation used here we refer the reader to [5,6]. Let G be a torsion-free group. We shall denote the typeset of G by T{G) = {type c (gf): 0 =/ = g e G}. Recall that a subgroup H of G is said to be balanced in G if H is pure in G and, for every g e G, there exists h e H such that the p-height of g + h in G is equal to the p-height of g + H in G/H for every prime p. This is equivalent to saying that the sequence CO ^ XH(T,P) = XHW. * = 1.2.