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1999 | Buch

Algebra

Some Recent Advances

herausgegeben von: I. B. S. Passi

Verlag: Hindustan Book Agency

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Inhaltsverzeichnis

Frontmatter
On Abelian Difference Sets
Abstract
We review some existence and nonexistence results — new and old — on abelian difference sets. Recent surveys on difference sets can be found in Arasu (1990), Jungnickel (1992a, b), Pott (1995), Jungnickel and Schmidt (1997), and Davis and Jedwab (1996). Standard references for difference sets are Baumert (1971), Beth et al. (1998), and Lander (1983). This article presents a flavour of the subject, by discussing some selected topics.
K. T. Arasu, Surinder K. Sehgal
Unit Groups of Group Rings
Abstract
For a commutative ring R with identity and an arbitrary group G, let RG denote the group ring of G over R and U(RG) its group of units. It is of interest, see the survey by Dennis (1977), to determine the necessary and sufficient conditions on R and G in order that U(RG) has a specific group-theoretic property, e.g., solvability, nilpotence, etc.. Considerable work has been done, by various authors, on these questions over the last thirty years or so. Besides solvability and nilpotence, other group-theoretic problems for the unit groups of group rings like the characterization of residual nilpotence, residual solvability, being an FC-group, torsion elements forming a subgroup, as well as the behaviour of the upper central series of the unit groups have also been studied. S.K. Sehgal’s book (1978) covers the main results obtained in this direction up to 1977 while the article by C. Polcino Milies (1981) gives some later developments (see also the books of Sehgal, 1989 and Karpilovsky, 1989). In this article our main aim is to survey the more recent developments. In §1 we review the case when R is a field and in §2 the case of the integral group ring is considered.
Ashwani K. Bhandari, I. B. S. Passi
Projective Modules Over Polynomial Rings
Abstract
In 1976, Quillen [Q] and Suslin [Su 1] proved the following conjecture of Serre:-
Conjecture: (Serre) Every finitely generated projective module over a polynomial ring k[T1,…, T n ] over a field k is free.
S. M. Bhatwadekar
Around Automorphisms of Relatively Free Groups
Abstract
Let V (≥ F′) be a fully invariant subgroup of a free group F = 〈x1,…, x n of rank n ≥ 2. A system w(V) = (w1 V,…, w m V), m ≤ n, of cosets of F / V is said to be primitive in F / V if it can be extended to a basis of F / V; and if w(V) is a primitive system in F / V then w(V) is said to lift to a primitive system of the free group F if there exists v = (v1,…, v m ), v i V, such that the system w(v) = (w1v1, …. , w m v m ) is primitive in F. Primitivity of a system w(V) = (w1 V,…, w n V), with n = rank (F), defines an automorphism α : F / VF / V mapping x i V → w i V, i = 1,…, n. Clearly, every automorphism, α : FF induces an automorphism α : F / VF / V (α ∈ Aut (F / V)). An automorphism α ∈ Aut (F / V) is said to be tame if it is induced by an automorphism α : FF (α ∈ Aut (F)) of the free group; or equivalently, if the corresponding primitive system w(V) = (w1 V,…, w n V) of cosets in F / V lifts to a primitive system w(v) = (w1v1,…, w n v n ) in F. If α ∈ Aut (F / V) is not tame then α may also be called wild or simply non-tame. Thus, for instance, if F is free of rank n ≥ 4, then every automorphism of F / F″ is tame (Bachmuth & Mochizuki, 1985) and consequently, every primitive system mod F″ lifts to a primitive system of F. However, in the case when n = 3, while the primitive system (x[y, z, x, x], y, z) mod F″ does not lift to a primitve system of F (Chein, 1968); the primitive element x[y, z, x, x]F″ is the same as x[x, [y, z, x]−1]F″ which in turn lifts to a part of the basis {x[x, [y, z, x]−1], y[y, [y, z, x]−1], z[z, [y, z, x]−1]} of F.
C. K. Gupta
Jordan Decomposition
Abstract
The Jordan canonical form for matrices over algebraically closed fields is standard fare in many linear algebra courses. The Jordan decomposition (into semisimple and nilpotent parts) for matrices over perfect fields is perhaps less well known, though very useful in many areas and closely related to the canonical form. This Jordan decomposition extends readily to elements of group algebras over perfect fields. During the past decade or so there has been activity in extending the decomposition to group rings (and matrices) over integral domains. In this article, we give a survey of this recent work (Arora et al., 1993 & 1998; Hales et al., 1990 & 1991) as well as some background on the classical results.
A. W. Hales, I. B. S. Passi
On the Normalizer Problem
Abstract
The object of this note is the structure of the normalizer of a group basis of the group ring RG of a finite group G, where R = or more generally in the situation when R is G-adapted. This means that R is an integral domain of characteristic zero in which no prime divisor of |G| is invertible.
Wolfgang Kimmerle
Galois Cohomology of Classical Groups
Abstract
In this article, we survey recent results of Eva Bayer-Fluckiger and the author on the Galois cohomology of classical groups over fields of virtual cohomological dimension 2. Number fields are examples of such fields. We begin by describing a well-known classification theorem for quadratic forms over number fields in terms of the so-called classical invariants (§ 2). We explain in § 3 how this classification leads to Hasse principle for principal homogeneous spaces for Spin q over number fields. In § 4 and § 7, we state the conjecture of Serre concerning the triviality of principal homogeneous spaces under semi-simple, simply connected, linear algebraic groups over perfect fields of cohomological dimension 2 and its real analogue due to Colliot-Thélène and Scheiderer in the form of a Hasse principle, if the field has virtual cohomological dimension ≤ 2. As in the case of Spin q over number fields, a main step in the proof of these conjectures is a classification theorem of hermitian forms over involutorial division algebras defined over fields of virtual cohomological dimension ≤ 2, which is described in § 6 and § 7.
R. Parimala
Central Units in Integral Group Rings
Abstract
This paper is intended to give a survey of recent work on central units in integral group rings. For units in general, the definitive reference is the book by Sehgal (1993) while a survey paper by Jespers contains additional very recent results. Both of these sources contain results on central units (in fact, Jespers devotes a chapter to the topic), but our work complements theirs in two ways. Firstly, we describe some results contained in papers which were not available to the other authors. Secondly, we choose to emphasize some topics which are mentioned either very briefly or not at all in their work. Nevertheless, we acknowledge that there is considerable overlap, especially between our survey and that of Jespers, and would like to thank him for supplying us with a preprint.
M. M. Parmenter
Alternative Loop Rings and Related Topics
Abstract
Let R be a commutative (and associative) ring with unity and let L be a loop (roughly speaking, a loop is a group which is not necessarily associative, see Definition 3.1). The loop algebra of L over R was introduced in 1944 by R.H. Bruck (1944) as a means to obtain a family of examples of nonassociative algebras and is defined in a way similar to that of a group algebra; i.e., as the free R-module with basis L, with a multiplication induced distributively from the operation in L.
César Polcino Milies
L-values at Zero and the Galois Structure of Global Units
Abstract
This article intends to present a comprehensive survey of the striking interplay between the Galois structure of the group of units in a number field and the values at zero of Artin L-functions. The algebraic ingredients come from integral representation theory, the ones from number theory include the Main Conjecture of Iwasawa theory. In fact, the discussion of recently defined invariants which go along with the unit group seems to propose possible generalizations of the Main Conjecture and fits very well into the framework of rather general conjectures regarding L-values by providing first affirmative answers. To begin with, we collect the principal ideas.
Jürgen Ritter
A Survey of Groups in Which Normality or Permutability is a Transitive Relation
Abstract
In group theory it is a familiar observation that normality of subgroups is not in general a transitive relation, i.e., HKG need not imply that HG. The smallest group exhibiting this phenomenon is Dih(8), the dihedral group of order 8.
Derek J. S. Robinson
The Structure of Some Group Rings
Abstract
For details of block theory we refer to [CR; 82, 6].
Klaus W. Roggenkamp
Symmetric Elements and Identities in Group Algebras
Abstract
Let FG be the group ring of a group G over a field F of characteristic p ≥ 0. Let * be the natural involution, γ = ∑γ(g)gγ* = ∑γ(g)g-1. Let us denote by
$${{\left( {KG} \right)}^{+}}=\left\{ {\gamma \in FG:\gamma *=\gamma } \right\}\text{and}{{\left( {KG} \right)}^{-}}=\left\{ {\gamma \in FG:\gamma *=-\gamma } \right\},$$
the sets of symmetric and skew symmetric elements respectively. We investigate whether certain identities on these and similar subsets control identities on the whole ring.
Sudarshan K. Sehgal
Serial Modules and Rings
Abstract
The fundamental theorem of abelian groups states that any finitely generated abelian group is a direct sum of cyclic groups. This theorem plays a fundamental role in the structure theory of abelian groups. It has fascinated many algebraists to look at this theorem from different points of view:-
(a)
To find suitable generalizations of this theorem for modules over certain classes of rings, e.g. Dedekind domain, hereditary noetherian prime rings, valuation rings etc.
 
(b)
To create suitable versions of this theorem in some module categories and use such versions to develop the structure theory of such module categories. For example, torsion abelian group-like modules, modules with finitely generated submodules direct sums of multiplication modules.
 
(c)
To find those rings for which certain versions of the fundamental theorem of abelian groups hold. For example rings over which all finitely generated modules are direct sums of cyclic modules.
 
(d)
To examine the structure of certain classes of abelian groups and to try to find modules with similar structures.
 
(e)
To find roles of answers to some of above types of questions in the general theory of modules.
 
Surjeet Singh
On Subgroups Determined by Ideals of an Integral Group Ring
Abstract
Let G be a group, ZG the integral group ring of G and I(G) its augmentation ideal. Recall that I(G) is the kernel of the ring homomorphism ∈: ZGZ given by ∈(∑n i g i ) = ∑n i , n i Z, g i G, and it is generated as a free Z-module by the elements g - 1, gG, ge. For n ≥ 1, I n (G) denote the nth associative power of I(G). For an ideal J of ZG, let G ∩ (1 + J) = {xG/x - 1 ∈ J}. Observe that for x, yG ∩ (1+ J), zG,
$$\begin{array}{*{20}{c}} {xy - 1 = x\left( {y - 1} \right) + x - 1 \in J} \\ {{x^{ - 1}} - 1 = - {x^{ - 1}}\left( {x - 1} \right) \in J,} \\ \end{array}$$
and
$${z^{ - 1}}xz - 1 = {z^{ - 1}}\left( {x - 1} \right)z \in J,$$
which imply that G ∩ (1 + J) is a normal subgroup of G. This subgroup is called the subgroup of G determined by the ideal J of ZG. When J = I n (G), n ≥ 1, the subgroup G ∩ (1 + I n (G)) = D n (G) is called the nth integral dimension subgroup of G and as been well studied during the last forty years—but we donot discuss this problem here (cf. Gupta, 1987; Gupta et al., 1984; Gupta and Kuzmin-unpublished; Passi et al., 1968, 1974, 1979, 1987, 1983; and Sandling 1972a & b; the list of references for dimension subgroups is by no means exhaustive).
L. R. Vermani
A Complex Irreducible Representation of the Quaternion Group and a Non-free Projective Module over the Polynomial Ring in Two Variables over the Real Quaternions
Abstract
It is a classical result of Frobenius and Schur ([F], p. 20–22 or [Se], p. 121–124) that any finite dimensional complex irreducible representation of a finite group, whose character is real, either descends to a real representation or can be extended to a representation of the group over the real quaternion algebra. The simplest example where the latter phenomenon holds is the standard 2-dimensional complex representation of the group of integral quaternions. An application of some general results of Barth and Hulek shows that this representation leads to a canonical rank 2 (stable) vector bundle over the complex projective plane. It can be shown that the restriction of this bundle to the affine plane gives rise to a non-free projective module of H [X, Y], isomorphic to the one constructed in [OS] in another context in a different manner. (The existence of this projective module led, incidentally, to the construction in ([P1]) of non diagonalisable, (in fact indecomposable), non singular symmetric 4×4 matrices of determinant one over the polynomial ring in two variables over the field of real numbers, producing remarkable counter examples to the so called quadratic analogue of Serre’s conjecture and opening up a new and fruitful area of research (cf. [P3]). On the other hand, it was shown in ([KPS]) that any non-free projective module over D[X, Y], where D is a finite dimensional division algebra over a field, extends (and essentially uniquely) to a vector bundle over the projective plane over this field, with a D-structure.
R. Sridharan
Metadaten
Titel
Algebra
herausgegeben von
I. B. S. Passi
Copyright-Jahr
1999
Verlag
Hindustan Book Agency
Electronic ISBN
978-93-80250-94-6
Print ISBN
978-81-85931-20-3
DOI
https://doi.org/10.1007/978-93-80250-94-6