Free Abelian Group Of Rank Research Articles

A NOTE ON NIELSEN EQUIVALENCE IN FINITELY GENERATED ABELIAN GROUPS

AbstractNielsen transformations determine the automorphisms of a free group of rank n, and also of a free abelian group of rank n, and furthermore the generating n-tuples of such groups form a single Nielsen equivalence class. For an arbitrary rank n group, the generating n-tuples may fall into several Nielsen classes. Diaconis and Graham [‘The graph of generating sets of an abelian group’, Colloq. Math.80 (1999), 31–38] determined the Nielsen classes for finite abelian groups. We extend their result to the case of infinite abelian groups.

CONGRUENCES OF L-VALUES FOR CYCLIC EXTENSIONS

Abstract. We study the consequences of Gross’s conjecture forcyclic extensions of degree l 2 where l is prime, and deduce that the L -values at s = 0 satisfy certain congruence relations. 1. IntroductionWe flrst review Gross’s conjecture brie°y. Let K=k be an abelianextension of global flelds with Galois group G . Let S be a flnite non-empty set of places of k which contains all archimedean places and allplaces ramifled in K , and let T be a flnite non-empty set of places of k which is disjoint from S . We choose T so that U S;T , the group of S -unitsin k which are congruent to 1 (mod v ) for all v 2 T , is a free abeliangroup of rank n = jSji 1.For a complex character ´ 2 G b = Hom( G; C ⁄ ), the associated modi-fled L -function is deflned as L S;T ( ´;s ) =Y v2T (1 i´ ( g v ) Nv 1 is )Y v62S (1 i´ ( g v ) Nv is ) i 1 ; where g v 2 G is the Frobenius element for v . The Stickelberger element µ G 2 C[ G ] is the unique element that satisfles ´ ( µ G ) = L S;T ( ´; 0)for all ´ 2 G b. In fact,

A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV .F or ag ivenn ∈ N ∪ {ω}, these groups of germs are free abelian groups of rank r ,f orr ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin's theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associ- ated groups of germs. Thus, if m �= n the groups mV and nV cannot be isomorphic. This answers a question of Brin.

Explicit Elliptic $K3$ Surfaces with Rank 15

This note presents a relatively straightforward proof of the fact that, under certain congruence conditions on a, b, c is an element of Q, the group of rational points over (Q) over bar (t) on the elliptic curve given by y(2) = x(3) + t(3)(t(2) + at + b)(2)(t + c)x + t(5)(t(2) + at + b)(3) is trivial. This is used to show that a related elliptic curve yields a free abelian group of rank 15 as its group of (Q) over bar (t)-rational points.

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

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.

Groups generated by 3-state automata over a 2-letter alphabet. II

This is the second in a series of papers presenting results on the classification of groups generated by 3-state automata over a 2-letter alphabet. Among the examples treated here, one can find automata generating the free product of 3 cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups, and other interesting self-similar groups.

Some remarks on generalized roundness

By using the links between generalized roundness, negative type inequalities and equivariant Hilbert space compressions, we obtain that the generalized roundness of the usual Cayley graph of finitely generated free groups and free abelian groups of rank ≥ 2 equals 1. This answers a question of J-F. Lafont and S. Prassidis.

Finite index supergroups and subgroups of torsionfree abelian groups of rank two

Every torsionfree abelian group A of rank two is a subgroup of Q ⊕ Q and is expressed by a direct limit of free abelian groups of rank two with lower diagonal integer-valued 2 × 2 -matrices as the bonding maps. Using these direct systems we classify all subgroups of Q ⊕ Q which are finite index supergroups of A or finite index subgroups of A. Using this classification we prove that for each prime p there exists a torsionfree abelian group A satisfying the following, where A ⩽ Q ⊕ Q and all supergroups are subgroups of Q ⊕ Q : (1) for each natural number s there are ∑ q | s , gcd ( p , q ) = 1 q s-index supergroups and also ∑ q | s , gcd ( p , q ) = 1 q s-index subgroups; (2) each pair of distinct s-index supergroups are non-isomorphic and each pair of distinct s-index subgroups are non-isomorphic.

Groups generated by 3-state automata over a 2-letter alphabet, I

This is the second in a series of papers presenting results on the classification of groups generated by 3-state automata over a 2-letter alphabet. Among the examples treated here, one can find automata generating the free product of 3 cyclic groups of order 2, a virtually free abelian group of rank 3, a solvable group of derived length 3, some virtually torsion-free weakly branch groups, and other interesting self-similar groups.

Test Ranks of Free Nilpotent Groups

ABSTRACT The test rank tr(G) of G is the minimum cardinality of a test set. In this paper we prove: I. Let N = N rc be a free nilpotent group of rank r ≥ 2 and class c ≥ 2. Then (i) tr(N) = 2 for r odd and c = 2; (ii) tr(N) = 1 in all other cases; (iii) an element g ∈ N 2q, 2 is a test element if and only if it can be written as g = ⋅ s , where t 1,…, t q are nonzero integers. II. Let F r be a free group of rank r ≥ 2 and A n be a free abelian group of rank n ≥ 1. Then the group G = G rn = F r ×A n has the test rank n. #Communicated by E. zelmanov.

AUTOMORPHISMS OF PRIME ORDER OF FREE METABELIAN GROUPS

ABSTRACT Let be a prime integer and let be a free metabelian group of rank In the paper an automorphism of order permuting cyclically fixed bases of is studied. The subgroup of fixed points of and the properties of action of on the commutator subgroup of are described. There are also given some results concerning such automorphisms of free abelian groups of rank

A Presentation of the Automorphism Group of the Two-Generator Free Metabelian and Nilpotent Group of Class c

We determine the structure of IA(G)/Inn(G) by giving a set of generators, and showing that IA(G)/Inn(G) is a free abelian group of rank (c − 2)(c + 3)/2. Here G = M2, c = 〈 x, y〉, c ≥ 2, is the free metabelian nilpotent group of class c.

STABILITY THEOREMS FOR SPACES OF RATIONAL CURVES

on the space of smooth based maps Map(Σ, X), would encode a lot of the topology of the space in terms of the critical points of the functional. These critical points are harmonic maps; the absolute minima, in particular, are often of special interest. It has been known, however, at least since the work of Sachs-Uhlenbeck that the Palais-Smale condition C fails for this functional, making a good Morse theory impossible. Nevertheless, in some cases, there is a sense in which the conclusions of Morse theory seem to hold, at least asymptotically. For simplicity, let us suppose that π1(X) = 0, and that π2(X) is a free abelian group of rank r so that the homotopy class of any map f ∈ Map(Σ, X) is given by a multi-degree k. We can try to compare the homology (homotopy) groups of the space of absolute minima Mink(Σ, X) with the homology (homotopy) groups of the full mapping space Mapk(Σ, X). If X is Kahler, then by a theorem of Eells and Wood [EW] the space of absolute minima Mink(Σ, X) is just the space Holk(Σ, X) of based holomorphic maps (or anti-holomorphic maps, depending on orientation) from Σ to X, as long as Holk(Σ, X) is non-empty. We are then led to consider the inclusion

Non-arithmetic polycyclic groups

We prove that for every n > 2 there are semi-direct products of a free Abelian group of rank n and an infinite cyclic group which are not arithmetic groups. In particular, we show that there are non-arithmetic polycyclic groups of every Hirsch number > 3. We also show that all polycyclic groups of Hirsch number ≤ are arithmetic. The proofs are based on our previous results about solvable arithmetic groups and on arithmetic properties of algebraic tori.

Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

We study the problem of finite approximability with respect to conjugacy of amalgamated free products by a normal subgroup and prove the following assertions. A) IfG is the amalgamated free productG=G1*HG2 of polycyclic groupsG1 andG2 by a normal subgroupH, whereH is an almost free Abelian group of rank 2, thenG is finitely approximate with respect to conjugacy. B) (i) IfG1=G2=L is a polycyclic group andG=G1*HG2 is the amalgamated product of two copies of the groupL by a normal subgroupH, thenG is finitely approximable with respect to conjugacy. (ii) IfG is an amalgamated free productG=G1*HG2 of polycyclic groupsG1 andG2 by a normal subgroupH, whereH is central inG1 orG2, thenG is finitely approximable with respect to conjugacy.

On the global dimension of multiplicative Weyl algebras

0. Introduction. Let k be any field and let k [x~ 1 . . . . . x + i] be the Laurent polynomial ring in n indeterminates. It is i somorphic to the group ring kG where G is the free abelian group of rank n. Given an element ~ ~ H2(G, k*) (G acting trivially on k*) we can construct the twisted group ring k" G. As a k-vector space, it is i somorphic to the group ring kG and ff {u~} ,~ is a basis, we define the mult ipl icat ion by the rule uxi uxj = f ( x l , x i )u~x j where f : G x G ~ k* is a 2-cocycle representing ~. An element ~ H 2 (G, k*) can be interpreted as a system of (~)-commutators

The Discrete Elementary Groups

This paper discusses discrete elementary subgroups in the Mobius group M( R n ).With a help of geometry,it is proved that these groups are isomorphic to either a group extension of Z by a finite subgroup of SO ( n ) or an extension of a finite group by a free Abelian group of rank k ≤ n .

Garden of Eden Configurations for Cellular Automata on Cayley Graphs of Groups

The tessellation of the plane given by square cells of equal size can be considered the Cayley graph of the free abelian group of rank 2. This group has polynomial growth. The theorems of Moore [Symposium on Applied Mathematics, Vol. XIV, American Mathematical Society, Providence, Rhode Island, 1962, pp. 17–33] and Myhill [Proceedings of the American Mathematical Society, 14 (1963), pp. 685–686] on the existence of Garden of Eden configurations for an automaton defined on such a graph are extended to Cayley graphs of groups whose growth function is not exponential. Examples are given of Cayley graphs of groups of exponential growth for which these theorems do not hold.

Sharply transitive posets with free abelian automorphism group

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.

A generalized cyclotomic identity

In this paper we prove a generalization of the cyclotomic identity for each group G satisfying suitable finiteness conditions. We prove this generalized cyclotomic identity by counting coloured G-set structures. The cyclotomic identity corresponds to taking G the free abelian group of rank one. For the free abelian group of rank two, for example, we get an analogous identity involving the generating function of the partition function. Interalia we show that studying G-set structures gives a unified and conceptually simple approach to other classical combinatorial identities not, to the author's knowledge, previously linked in the literature.