Group Of Rational Numbers Research Articles

ON A QUESTION OF A. SOZUTOV

In the paper an answer to a problem posed by A.I. Sozutov in the Kourovka Notebook is given. The solution is based on some modification of the method that was proposed for constructing a non-abelian analogue of the additive group of rational numbers, i.e. a group whose center is an infinite cyclic group and any two non-trivial subgroups of which have a non-trivial intersection.

Douglas Algebras on the Boundary of the Big Disk

Let G be the compact group of all characters of the additive group of rational numbers, and let H ∞ G be the Banach algebra of so called bounded hyper- analytic functions on the big-diskG. We give a sufficient condition for an algebra between H ∞ G and L ∞ be a Douglas algebra.

Divisibility of reduction in groups of rational numbers

Given a multiplicative group of nonzero rational numbers and a positive integer m m , we consider the problem of determining the density of the set of primes p p for which the order of the reduction modulo p p of the group is divisible by m m . In the case when the group is finitely generated the density is explicitly computed. Some examples of groups with infinite rank are considered.

Dominions in abelian subgroups of metabelian groups

It is proved that a suitable free Abelian group of finite rank is not absolutely closed in the class $ {\mathcal A} $ 2 of metabelian groups. A condition is specified under which a torsion-free Abelian group is not absolutely closed in $ {\mathcal A} $ 2. Also we gain insight into the question when the dominion in $ {\mathcal A} $ 2 of the additive group of rational numbers coincides with this subgroup.

A problem from the kourovka notebook on embeddings of the group $$ \mathbb{Q} $$

Answering the question of de la Harpe and Bridson in the Kourovka Notebook, Problem 14.10(b), we construct explicit embeddings of the additive group of rational numbers \( \mathbb{Q} \) in a finitely generated group G. In fact, the group G is 2-generator, and the constructed embedding can be subnormal and preserve a few properties such as solubility or torsion freeness.

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

In [Todor Tsankov, The additive group of the rationals does not have an automatic presentation, May 2009, arXiv:0905.1505v1], it was shown that the group of rational numbers is not FA-presentable, i.e. it does not admit a presentation by a finite automaton. More generally, any torsion-free abelian group that is divisible by infinitely many primes is not of this kind. In this article we extend the result from [13] and prove that any torsion-free FA-presentable abelian group G is an extension of a finite rank free group by a finite direct sum of Prüfer groups ℤ(p∞).

A characterization of compactly generated metric groups

Recall that a topological group $G$ is: (a) $\sigma$-compact if $G=\bigcup \{K_n:n\in \mathbf N\}$ where each $K_n$ is compact, and (b) compactly generated if $G$ is algebraically generated by some compact subset of $G$. Compactly generated groups are $\sigma$-compact, but the converse is not true: every countable non-finitely generated discrete group (for example, the group of rational numbers or the free (Abelian) group with a countable infinite set of generators) is a counterexample. We prove that a metric group $G$ is compactly generated if and only if $G$ is $\sigma$-compact and for every open subgroup $H$ of $G$ there exists a finite set $F$ such that $F\cup H$ algebraically generates $G$. As a corollary, we obtain that a $\sigma$-compact metric group $G$ is compactly generated provided that one of the following conditions holds: (i) $G$ has no proper open subgroups, (ii) $G$ is dense in some connected group (in particular, if $G$ is connected itself), (iii) $G$ is totally bounded (= subgroup of a compact group). Our second major result states that a countable metric group is compactly generated if and only if it can be generated by a sequence converging to its identity element (eventually constant sequences are not excluded here). Therefore, a countable metric group $G$ can be generated by a (possibly eventually constant) sequence converging to its identity element in each of the cases (i), (ii) and (iii) above. Examples demonstrating that various conditions cannot be omitted or relaxed are constructed. In particular, we exhibit a countable totally bounded group which is not compactly generated.

A BANACH SPACE WHOSE OPERATOR ALGEBRA HAS K0-GROUP ℚ

We construct a Banach space X whose algebra of bounded linear operators has K0-group the additive group of rational numbers. We achieve this using a Gowers-Maurey space.2000 Mathematical Subject Classification: 20F36, 55P35, 55Q05, 55Q40, 55U10

Tight subgroups in almost completely decomposable groups

A completely decomposable group is a direct sum of groups isomorphic to subgroups of the additive group of rational numbers. An almost completely decomposable group X is a finite essential (abelian) extension of a completely decomposable group A of finite rank. In 1974, Lady [Lad74] initiated a systematic theory of such groups based on the fundamental concept of regulating subgroup. The regulating subgroups can be defined as the completely decomposable subgroups of least index in an almost completely decomposable group. Details on the subsequent developments can be found in the survey article [Mad95] or in the monograph [Mad99].

Absolute decomposability of the group of rational numbers

Absolute decomposability of the group of rational numbers

On quasi-decompositions of B(1)-groups

Suppose G is a torsion-free abelian group of finite rank which is quasi-isomorphic to G1⊕…⊕Gk with each Gi strongly indecomposable. Call G rigid if Hom(Gi,Gj) = 0 for all i≠j. If A1,…,An are nonzero subgroups of the additive group of rational numbers, G[A1,…,An] denotes the cokernel of the diagonal map ⋂Ai→⊕Ai In this paper, a study of rigid groups of the form G[A1,…,An] is initiated

The semigroup of nonempty finite subsets of rationals

Let Q be the additive group of rational numbers and let ℛ be the additive semigroup of all nonempty finite subsets of Q. For X ∈ ℛ, define AX to be the basis of 〈X − min(X)〉 and BX the basis of 〈max(X) − X〉. In the greatest semilattice decomposition of ℛ, let 𝒜(X) denote the archimedean component containing X. In this paper we examine the structure of ℛ and determine its greatest semilattice decomposition. In particular, we show that for X, Y ∈ ℛ, 𝒜(X) = 𝒜(Y) if and only if AX = AY and BX = BY. Furthermore, if X is a non‐singleton, then the idempotent‐free 𝒜(X) is isomorphic to the direct product of a power joined subsemigroup and the group Q.

64.27 Isomorphisms between groups of rational numbers

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Tensor products of mixed abelian groups

For any abelian group G, let tG denote the maximal torsion subgroup of G. Group G is said to be mixed if 0 # tG f: G and to be of rank one if G/tG is isomorphic to a subgroup of Q, the additive group of rational numbers. In [4], it is shown that a countable, rank-one mixed group G with tG reduced is determined up to isomorphism by its Ulm invariants and an invariant U(G), which we shall discuss in more detail below; in other words, a countable rank-one abelian group G is determined by tG and U(G). If G and K are both countable rank-one mixed groups with reduced torsion parts, then their tensor product G @ K = G & K also is easily seen to be such a group, Thus, if we can determine the Ulm invariants of G @ K and compute the invariant U(G @ K), we shall have determined the structure of G @ K. Also, let us mention that due to the fact that taking tensor products of torsion groups radically simplifies their structure, our results to some extent will be applicable to uncountable rank-one groups. Furthermore, the techniques we develop in computing U(G @ K) will probably be crucial to any future study of the structure of G @K for arbitrary mixed groups G and 2% Finally, we remark that taking tensor products of mixed groups does, as one would anticipate, lead to groups of simpler structure. Nonetheless, the groups G @ K remain rich enough in structure to make their determination nontrivial. Let p be an arbitrary prime. We define polG for all ordinals 01 by

Extensions of free groups by torsion groups

Recently R. G. Swan [51 has proven the following theorem with no commutative assumptions whatsoever: If a torsion free group G contains a free subgroup of finite index, then G is free. Of course, if G is abelian, there is a trivial generalization to the case where G/F is assumed to be bounded with F free (that is, n(G/F) = 0 where n$O0GZ). However, in this note we show that Swan's theorem has an even more striking improvement than the one suggested above for G an abelian group. For example, G/F may be a subgroup of a totally projective p-group of arbitrary length. In the ensuing discussion all groups will be abelian and p will be an arbitrary but fixed prime. We denote the group of integers by Z and the group of rational numbers by Q. A cotorsion functor S (defined by Nunke in [3]) is one that has a representing sequence Z>-+M--->H where H is torsion such that, for a group G, SG= Image(Hom(M, G)--Hom(Z, G)_G) and if f:G--A then S( =f SG. We further use the notation Sp when H is a p-group (that is, Sp is a p-coprimary functor in the sense of Nunke) and call Sp reduced if H is reduced. Hill and Megibben [2] call a group G a weak S-projective if for any free resolution Fo >-4F---> G of G and any group X we have that 6x(S Hom(Fo, X)) =0 where

Rational cohomology operations and Massey products

Let Q be the group of rational numbers. Then H*(Q, n; Q) is either an exterior algebra or polynomial algebra on a class u of dimension n. By the Kunneth formula, if PX>=K(Q, nj), that is if P is a rational generalized Eilenberg-MacLane space (GEM), then every class in Hk(P; Q) is a polynomial on the fundamental classes {ui}. Thus every rational primary cohomology operation on (xi, ** *, x) can be written d) {Xj I} = ZXjxj+ Eyrzr when XjE Q and yr, zrEI Q). The decomposable term is the twofold matrix Massey product

Abelian Groups with a Vanishing Homology Group

In this paper, we wish to characterize those abelian groups whose integral homology groups vanish in some positive dimension. We obtain a complete characterization provided the dimension in which the homology vanishes is odd; in fact, we prove that the only abelian groups which possess a vanishing homology group in an odd dimension are, up to isomorphism, subgroups of Qn, where Q denotes the additive group of rational numbers. The case of vanishing in an even dimension is much more complicated. We exhibit a class of groups whose homology vanishes in even dimensions and is otherwise very nice, namely the subgroups of Q/Z, and then show that unless we impose further restrictions, there exist abelian groups which possess the homology of subgroups of Q/Z without being isomorphic to a subgroup of Q/Z.

The Entropy of an Automorphism of a Solenoidal Group

Let us denote by X a group of characters for the subgroup R of an additive group of rational numbers and by T its automorphism adjoint to the automorphism of the group R, which is given by multiplying by the irreducible fraction ${m / n}$. In this paper it is proved that the entropy of such an automorphism equals $\log (\max \{ |m|,n\} )$.