Sharply transitive posets with free abelian automorphism group

1991 ◽  
Vol 109 (3) ◽  
pp. 517-520
Author(s):  
Gerhard Behrendt
Keyword(s):  
Abelian Group ◽  
Automorphism Group ◽  
Ordered Set ◽  
Free Abelian Group ◽  
Additive Group ◽  
Full Description ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Totally Ordered Set ◽  
Sharply Transitive

Let (X,≤) be a partially ordered set (in short, a poset). The automorphism group Aut (X,≤) is the group of all permutations g of X such that x ≤ y if and only if xg ≤ yg for all x,y∈X. We say that (X,≤) is sharply transitive if Aut(X,≤) is sharply transitive on X, that is, for x,y∈X there exists a unique g∈Aut(X,≤) with y = xg. Sharply transitive totally ordered sets have been studied by Ohkuma[4, 5], Glass, Gurevich, Holland and Shelah [3] (see also [2] and [6]). Whereas the only countable sharply transitive totally ordered set is the set of integers, there are a great variety of countable sharply transitive posets. Amongst other results, in [1] the author showed that there are countably many non-isomorphic sharply transitive posets whose automorphism group is infinite cyclic (and also gave a full description of those), whereas there are 2N0 non-isomorphic sharply transitive posets whose automorphism group is isomorphic to the additive group of the rational numbers. This suggests also that one should consider the analogous problem for free abelian groups. The purpose of this note is to show that whenever G is a countable free abelian group then there exists a sharply transitive poset whose automorphism group is isomorphic to G, and that there are already 2N0 non-isomorphic sharply transitive posets whose automorphism group is the free abelian group of rank 2.

Cellular covers of ℵ1-free abelian groups

2015 ◽  
Vol 14 (10) ◽  
pp. 1550139 ◽  
Author(s):  
José L. Rodríguez ◽  
Lutz Strüngmann
Keyword(s):  
Abelian Group ◽  
Exact Sequence ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Countable Rank ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Exact Sequences ◽  
Torsion Free ◽  
Cellular Cover

In this paper, we first show that for every natural number n and every countable reduced cotorsion-free group K there is a short exact sequence [Formula: see text] such that the map G → H is a cellular cover over H and the rank of H is exactly n. In particular, the free abelian group of infinite countable rank is the kernel of a cellular exact sequence of co-rank 2 which answers an open problem from Rodríguez–Strüngmann [J. L. Rodríguez and L. Strüngmann, Mediterr. J. Math.6 (2010) 139–150]. Moreover, we give a new method to construct cellular exact sequences with prescribed torsion free kernels and cokernels. In particular we apply this method to the class of ℵ1-free abelian groups in order to complement results from the cited work and Göbel–Rodríguez–Strüngmann [R. Göbel, J. L. Rodríguez and L. Strüngmann, Fund. Math.217 (2012) 211–231].

BREAKING UP FINITE AUTOMATA PRESENTABLE TORSION-FREE ABELIAN GROUPS

2011 ◽  
Vol 21 (08) ◽  
pp. 1463-1472 ◽  
Author(s):  
GÁBOR BRAUN ◽  
LUTZ STRÜNGMANN
Keyword(s):  
Abelian Group ◽  
Finite Automaton ◽  
Free Abelian Group ◽  
Finite Automata ◽  
Additive Group ◽  
Rational Numbers ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

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 note on rank and direct decompositions of torsion-free Abelian groups. II

1969 ◽  
Vol 66 (2) ◽  
pp. 239-240 ◽  
Author(s):  
A. L. S. Corner
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Direct Sums ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Rank 2 ◽  
Direct Decompositions ◽  
Torsion Free

According to well-known theorems of Kaplansky and Baer–Kulikov–Kapla nsky–Fuchs (4, 2), the class of direct sums of countable Abelian groups and the class of direct sums of torsion-free Abelian groups of rank 1 are both closed under the formation of direct summands. In this note I give an example to show that the class of direct sums of torsion-free Abelian groups of finite rank does not share this closure property: more precisely, there exists a torsion-free Abelian group G which can be written both as a direct sum G = A⊕B of 2 indecomposable groups A, B of rank ℵ0 and as a direct sum G = ⊕n ε zCn of ℵ0 indecomposable groups Cn (nεZ) of rank 2, where Z is the set of all integers.

On the SL(2, C)-representation rings of free abelian groups

2018 ◽  
Vol 167 (02) ◽  
pp. 229-247
Author(s):  
TAKAO SATOH
Keyword(s):  
Abelian Group ◽  
Upper Bound ◽  
Automorphism Groups ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Groups ◽  
The Other ◽  
Subsequent Research ◽  
Representation Rings ◽  
Free Abelian Groups

AbstractIn this paper, we study “the ring of component functions” of SL(2, C)-representations of free abelian groups. This is a subsequent research of our previous work [11] for free groups. We introduce some descending filtration of the ring, and determine the structure of its graded quotients.Then we give two applications. In [30], we constructed the generalized Johnson homomorphisms. We give an upper bound on their images with the graded quotients. The other application is to construct a certain crossed homomorphisms of the automorphism groups of free groups. We show that our crossed homomorphism induces Morita's 1-cocycle defined in [22]. In other words, we give another construction of Morita's 1-cocyle with the SL(2, C)-representations of the free abelian group.

scholarly journals On Geometric Flats in the CAT(0) Realization of Coxeter Groups and Tits Buildings

2009 ◽  
Vol 61 (4) ◽  
pp. 740-761 ◽  
Author(s):  
Pierre-Emmanuel Caprace ◽  
Frédéric Haglund
Keyword(s):  
Abelian Group ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Free Abelian Group ◽  
Gromov Hyperbolic ◽  
Convex Cocompact ◽  
Rank 2 ◽  
Cocompact Subgroup ◽  
Special Case ◽  
Geometric Action

Abstract.Given a complete CAT(0) space X endowed with a geometric action of a group Ⲅ, it is known that if Ⲅ contains a free abelian group of rank n, then X contains a geometric flat of dimension n. We prove the converse of this statement in the special case where X is a convex subcomplex of the CAT(0) realization of a Coxeter group W, and Ⲅ is a subgroup of W. In particular a convex cocompact subgroup of a Coxeter group is Gromov-hyperbolic if and only if it does not contain a free abelian group of rank 2. Our result also provides an explicit control on geometric flats in the CAT(0) realization of arbitrary Tits buildings.

scholarly journals Type radicals and quasi-decompositions of torsion-free abelian groups

1992 ◽  
Vol 52 (2) ◽  
pp. 219-236 ◽  
Author(s):  
Oteo Mutzbauer
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Torsion Free Abelian Group ◽  
Direct Sum ◽  
Free Abelian Groups ◽  
Torsion Free

AbstractA composition sequence for a torsion-free abelian group A is an increasing sequenceof pure subgroups with rank 1 quotients and union A. Properties of A can be described by the sequence of types of these quotients. For example, if A is uniform, that is all the types in some sequence are equal, then A is complete decomposable if it is homogeneous. If A has finite rank and all permutations ofone of its type sequences can be realized, then A is quasi-isomorphic to a direct sum of uniform groups.

Torsion-free abelian groups with optimal Scott families

2018 ◽  
Vol 18 (01) ◽  
pp. 1850002
Author(s):  
Alexander G. Melnikov
Keyword(s):  
Abelian Group ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Symbolic Logic ◽  
Technical Result ◽  
Torsion Free Abelian Group ◽  
Scott Family ◽  
Free Abelian Groups ◽  
Torsion Free ◽  
Labeled Tree

We prove that for any computable successor ordinal of the form [Formula: see text] [Formula: see text] limit and [Formula: see text] there exists computable torsion-free abelian group [Formula: see text]TFAG[Formula: see text] that is relatively [Formula: see text] -categorical and not [Formula: see text] -categorical. Equivalently, for any such [Formula: see text] there exists a computable TFAG whose initial segments are uniformly described by [Formula: see text] infinitary computable formulae up to automorphism (i.e. it has a c.e. uniformly [Formula: see text]-Scott family), and there is no syntactically simpler (c.e.) family of formulae that would capture these orbits. As far as we know, the problem of finding such optimal examples of (relatively) [Formula: see text]-categorical TFAGs for arbitrarily large [Formula: see text] was first raised by Goncharov at least 10 years ago, but it has resisted solution (see e.g. Problem 7.1 in survey [Computable abelian groups, Bull. Symbolic Logic 20(3) (2014) 315–356]). As a byproduct of the proof, we introduce an effective functor that transforms a [Formula: see text]-computable worthy labeled tree (to be defined) into a computable TFAG. We expect that this technical result will find further applications not necessarily related to categoricity questions.

scholarly journals Hypertypes of torsion-free abelian groups of finite rank

1989 ◽  
Vol 39 (1) ◽  
pp. 21-24 ◽  
Author(s):  
H.P. Goeters ◽  
C. Vinsonhaler ◽  
W. Wickless
Keyword(s):  
Abelian Group ◽  
Finite Rank ◽  
Free Abelian Group ◽  
Abelian Groups ◽  
Free Subgroup ◽  
Torsion Free Abelian Group ◽  
Free Abelian Groups ◽  
Torsion Free

Let G be a torsion-free abelian group of finite rank n and let F be a full free subgroup of G. Then G/F is isomorphic to T1 ⊕ … ⊕ Tn, where T1 ⊆ T2 ⊆ … ⊆ Tn ⊆ ℚ/ℤ. It is well known that type T1 = inner type G and type Tn = outer type G. In this note we give two characterisations of type Ti for 1 < i < n.

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