Algebra of multirate structures

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.

Abelian torsion-free groups of finite rank with finite automorphism groups are considered as rigid extensions of a system of strongly indecomposable groups Aj, j = 1, ..., k, of finite rank and having finite automorphism groups, by a finite p-group P. Such groups are called (A, p)-groups. The author introduces for (A, P)-groups the concept of (A, P)-type, which represents a choice of k integer matrices. A complete description of (A, P)-groups is given by means of (A, P)-types. Using this description, a series of problems on finite groups of automorphisms of torsion-free abelian groups of finite rank are solved. Furthermore, it is shown that the actual solution of any one of these problems comes down to a question of the consistency of a system of equations of the first degree modulo pt, where pt is the maximal order of elements of P. Bibliography: 11 titles.

A BV-algebra structure on Hochschild cohomology of the integral group ring of finitely generated Abelian groups

I have developed an algorithm and written a GAP functions pautfreegroup:=function(n) (Finite Presentation of automorphism Groups of Free group of Rank n), and autfreeabeliangroup:=function(n) (Finite Presentation of automorphism Groups of Free abelian group of Rank n). Functions using GAP system for computation of a finite presentation for Aut(F_n ) and GL(n,z) respectively. In order to do that we have given a description for the presentation of Aut(F_n) and GL(n,z), the automorphism groups of free groups and free abelian groups respectively.

Quasi-pure projective and injective torsion free abelian groups of rank 2

The primary purpose of this chapter is the analysis of the group Wq(R) in the case of an arithmetic Dedekind domain R. The characterization of the elements of Br(R)2 as the Brauer classes of Clifford algebras of quadratic spaces over R of rank 4 and trivial Arf invariant is an important step along the way. The quadratic spaces which it provides make it possible to describe Ker Arf by use of the total signature. In the special case where R is the ring of integers in a number field, the quadratic group Qu(R) is closely related to the ideal class group, and this connection together with the description of Ker Arf implies that Wq(R) ≅ Cl(R)2 ⊕ G, where C1(R) is the ideal class group of R and G is a free Abelian group of rank with r the number of real embeddings of the number field. An additional focus is the comparison of the number theory of Wq(R) with that of W(R) and the structure of the quotient W(R)/Wq(R).

Given a free Abelian group of finite rank, canonical subgroup matrices are defined with respect to an arbitrary but fixed basis of , which uniquely determine all subgroups of of finite index. The use of these matrices allows the classification of all subgroups of fixed finite index. An application is made in the construction of the intersection group and the union group of two arbitrary subgroups of finite index and upper and lower bounds on the index of these groups are found. Physical applications occur for rank three in the systematic symmetry analysis of domain structures, which appear in structural phase transitions in solids.

Multiple zeta functions and asymptotic structure of free Abelian groups of finite rank

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.

We define a one-to-one correspondence between torsion-free Abelian groups of finite rank with marked bases and finite sequences of elements of finitely presented modules over the ring of polyadic numbers. This correspondence is a duality of two categories.

In this paper, we shall discuss some recent contributions to the project [15, 14, 2, 18, 22, 23] of explaining why no satisfactory system of complete invariants has yet been found for the torsion-free abelian groups of finite rank n ≥ 2. Recall that, up to isomorphism, the torsion-free abelian groups of rank n are exactly the additive subgroups of the n-dimensional vector space ℚn which contain n linearly independent elements. Thus the collection of torsion-free abelian groups of rank at most n can be naturally identified with the set S (ℚn) of all nontrivial additive subgroups of ℚn. In 1937, Baer [4] solved the classification problem for the class S(ℚ)of rank 1 groups as follows.Let ℙ be the set of primes. If G is a torsion-free abelian group and 0 ≠ x ϵ G, then the p-height of x is defined to behx(p) = sup{n ϵ ℕ ∣ There exists y ϵ G such that pny = x} ϵ ℕ ∪{∞}; and the characteristic χ (x) of x is defined to be the function

The structure of the connected component of the identity of the idele-class group of a number field is given in Artin-Tate [1]. It seems worthwhile to investigate whether this result is the trivial of a theorem in the arithmetic of algebraic tori, i.e. the case corresponding to the multiplicative group Gm (as is true of the unit theorem and the finiteness of the class number). It turns out that the formalism of the theory of tori defined over number fields allows one to generalize Artin's proof line-by-line, obtaining the structure of the identity component of the adele-class group of any such torus. I am grateful to Professor T. Ono for suggesting the problem to me. I give now a summary of the definitions and results that I shall need (for all this, see Ono [3]). I denote by Z, Q, R, and C the rings of respectively integers, rationals, real numbers, and complex numbers, and by T an algebraic torus defined over a number field k. T is the group of rational characters of T. If the dimension of T is d, then T is a free abelian group of rank d [3, ?1.1]. T is split by a finite extension K of k, which I choose to be normal over Q. Then the group TK of points of T rational over K is isomorphic to the product (K*)d of K* with itself d times. Every & 'T is a rational map defined over K; hence if g denotes the galois group of K over k, T has the structure of a g-module. The group Hom(Ti, K*) has the structure of a gmodule, given byfo() = (f(a)), and we have g-isomorphisms

If a knot K has Alexander polynomial different from 1, then its knot group, G maps onto some metacyclic group, Zr§Zp. We show that in that case, it also has a homomorphism onto a split extension of a free abelian group of rank p — 1 by Zr§Zp, and hence also onto a split extension of a direct sum of p — 1 cyclic groups of order s by the metacyclic group. In many cases, (such as if s is coprime with p), this group can be specified exactly. Otherwise there are a finite number of possibilities. A special case is Perko's result that a homomorphism of a knot group onto S3 = Z2^Z?) lifts to S4 = Z 2ίS)Z3<$(Z2 Θ Z2). As an application we obtain information about the derived series of G. In a final section it is shown how to associate a rational polynomial invariant to every metacyclic representation. 1. Lifting metacyclic representations. Let p be a prime, r a divisor of p — 1 and β a primitive rth root modulo p. Let E — Zj§Zp — (Y, S: γr zn sp = 19Y'1SY= Sβ). Up to isomorphism, the group is independent of β. Let G be the knot group of a knot, AT, in the 3-sphere, S3, and let φ be a homomoφhism of G onto E which takes a meridian, m, of K to YaSh. Then g.c.d. (α, r) — 1, since G is generated by conjugates of m. Setting X — YaSb and eliminating 7, we obtain a presentation

It is proved that ifr1,r2, ...,rs;l1,l2, ...,lt are the ranks of the indecomposable summands of two direct decompositions of a torsion-free Abelian group of finite rank and if s0 is the number of units among the numbers ri, while t0 is the number of units among the numbers lj, thenri ⩽n - t0,lj⩽n−s0 for all i, j. Moreover, if for some i we have ri=n−t0, then among the lj's only one term is different from 1 and it is equal to n−t0; similarly if lj=n−s0 for some j. In addition, a construction is presented, allowing to form, from several indecomposable groups, a new group, called a flower group, and it is proved that a flower group is indecomposable under natural restrictions on its defining parameters.

We study Dirichlet series enumerating orbits of Cartesian products of maps whose orbit distributions are modelled on the distributions of finite index subgroups of free abelian groups of finite rank. We interpret Euler factors of such orbit Dirichlet series in terms of generating polynomials for statistics on multiset permutations, viz. descent and major index, generalizing Carlitz's $q$-Eulerian polynomials. We give two main applications of this combinatorial interpretation. Firstly, we establish local functional equations for the Euler factors of the orbit Dirichlet series under consideration. Secondly, we determine these (global) Dirichlet series' abscissae of convergence and establish some meromorphic continuation beyond these abscissae. As a corollary, we describe the asymptotics of the relevant orbit growth sequences. For Cartesian products of more than two maps we establish a natural boundary for meromorphic continuation. For products of two maps, we prove the existence of such a natural boundary subject to a combinatorial conjecture.