Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Konstruktion symmetrischer Designs

In dieser Arbeit werden zehn neue symmetrische (176,50,14) Designs und ein neues symmetrisches (144,66,30) Design durch Vorgabe von nichtauflösbaren Automorphismengruppen konstruiert. Im Jahre 1969 entdeckte G. Higman ein symmetrisches (176,50,14) Design, dessen volle Automorphismengruppe die sporadische einfache Gruppe HS der Ordnung 44.352.000 ist. Hier wurden nun Designs gesucht, die eine Untergruppe von HS zulassen. Folgende Untergruppen wurden betrachtet: die transitive und die intransitive Erweiterung einer elementarabelschen Gruppe der Ordnung 16 durch Alt(5), AGL(3,2), das direkte Produkt einer zyklischen Gruppe der Ordnung 5 mit Alt(5) und PSL(2,11). Die transitive Erweiterung von E(16) durch Alt(5) lieferte zwei neue Designs mit Automorphismengruppen der Ordnungen 960 bzw. 11.520; letzteres konnte auch mit der transitiven Erweiterung erhalten werden. Die Gruppe PSL(2,11) operiert auf den Punkten des Higman-Designs in drei Bahnen; sucht man nach symmetrischen (176,50,14) Designs, auf denen diese Gruppe in zwei Bahnen operiert, so erhält man acht neue Designs. Die übrigen Gruppen lieferten keine neuen Designs. Schließlich konnte ein neues symmetrisches (144,66,30) Design unter Verwendung der sporadischen Mathieu-Gruppe M(12) konstruiert werden. Dies war zu diesem Zeitpunkt außer dem Higman-Design das einzige bekannte symmetrische Design, dessen volle Automorphismengruppe im Wesentlichen eine sporadische einfache Gruppe ist.

A Simple Algorithm for Unique Representation of Chemical Structures - Cyclic/ Acyclic Functionalized Achiral Hydrocarbons

An algorithm, based on ‘vertex priority values’ has been proposed to uniquely sequence and represent connectivity matrix of chemical structures of cyclic/ acyclic functionalized achiral hydrocarbons and their derivatives. In this method ‘vertex priority values’ have been assigned in terms of atomic weights, subgraph lengths, loops, and heteroatom contents. Subsequently the terminal vertices have been considered upon completing the sequencing of the core vertices. This approach provides a multilayered connectivity graph, which can be put to use in comparing two or more structures or parts thereof for any given purpose. Furthermore the basic vertex connection tables generated here are useful in the computation of characteristic matrices/ topological indices, automorphism groups, and in storing, sorting and retrieving of chemical structures from databases.

Flexibility Properties in Complex Analysis and Affine Algebraic Geometry

In the last decades affine algebraic varieties and Stein manifolds with big (infinite-dimensional) automorphism groups have been intensively studied. Several notions expressing that the automorphisms group is big have been proposed. All of them imply that the manifold in question is an Oka–Forstnerič manifold. This important notion has also recently merged from the intensive studies around the homotopy principle in Complex Analysis. This homotopy principle, which goes back to the 1930s, has had an enormous impact on the development of the area of Several Complex Variables and the number of its applications is constantly growing. In this overview chapter we present three classes of properties: (1) density property, (2) flexibility, and (3) Oka–Forstnerič. For each class we give the relevant definitions, its most significant features and explain the known implications between all these properties. Many difficult mathematical problems could be solved by applying the developed theory, we indicate some of the most spectacular ones.

What is Q-extension?

In this paper we present a developed software in the area of Coding Theory. Using it, codes with given properties can be classified. A part of this software can be used also for investigations (isomorphisms, automorphism groups) of other discrete structures-combinatorial designs, Hadamard matrices, bipartite graphs etc.

New Symmetric (61,16,4) Designs Invariant Under the Dihedral Group of Order 10

∗ This work has been partially supported by the Bulgarian NSF under Contract No. I-506/1995.

Some generalization of Desargues and Veronese configurations

2000 Mathematics Subject Classification: 51E14, 51E30.

Subvarieties of the Hyperelliptic Moduli Determined by Group Actions

2000 Mathematics Subject Classification: 14Q05, 14Q15, 14R20, 14D22.

Relação de alcance em dígrafos transitivos e a propriedade Z

Dissertação (mestrado)—Universidade de Brasília, Instituto de Ciências Exatas, Departamento de Matemática, 2016.

Zero-sum problems in finite cyclic groups

The purpose of this thesis is to investigate some open problems in the area of combinatorial number theory referred to as zero-sum theory. A zero-sequence in a finite cyclic group G is said to have the basic property if it is equivalent under group automorphism to one which has sum precisely IGI when this sum is viewed as an integer. This thesis investigates two major problems, the first of which is referred to as the basic pair problem. This problem seeks to determine conditions for which every zero-sequence of a given length in a finite abelian group has the basic property. We resolve an open problem regarding basic pairs in cyclic groups by demonstrating that every sequence of length four in Zp has the basic property, and we conjecture on the complete solution of this problem. The second problem is a 1988 conjecture of Kleitman and Lemke, part of which claims that every sequence of length n in Zn has a subsequence with the basic property. If one considers the special case where n is an odd integer we believe this conjecture to hold true. We verify this is the case for all prime integers less than 40, and all odd integers less than 26. In addition, we resolve the Kleitman-Lemke conjecture for general n in the negative. That is, we demonstrate a sequence in any finite abelian group isomorphic to Z2p (for p ~ 11 a prime) containing no subsequence with the basic property. These results, as well as the results found along the way, contribute to many other problems in zero-sum theory.

Twisted conjugacy classes in nilpotent groups

A group is said to have the R(infinity) property if every automorphism has an infinite number of twisted conjugacy classes. We study the question whether G has the R(infinity) property when G is a finitely generated torsion-free nilpotent group. As a consequence, we show that for every positive integer n >= 5, there is a compact nilmanifold of dimension n on which every homeomorphism is isotopic to a fixed point free homeomorphism. As a by-product, we give a purely group theoretic proof that the free group on two generators has the R(infinity) property. The R(infinity) property for virtually abelian and for C-nilpotent groups are also discussed.

On automorphisms of split metacyclic groups

Let D( m, n; k) be the semi-direct product of two finite cyclic groups Z/m = < x > and Z/n = < y >, where the action is given by yxy(-1) = x(k). In particular, this includes the dihedral groups D(2m). We calculate the automorphism group Aut (D(m, n; k)).

On cohomologies and extensions of cyclic groups

Cohomology groups H(s)(Z(n), Z(m)) are studied to describe all groups up to isomorphism which are (central) extensions of the cyclic group Z(n) by the Z(n)-module Z(m). Further, for each such a group the number of non-equivalent extensions is determined. (C) 2011 Elsevier B.V. All rights reserved.

A NOTE ON FREE GROUPS IN THE RING OF FRACTIONS OF SKEW POLYNOMIAL RINGS

Let L be a function field over the rationals and let D denote the skew field of fractions of L[t; sigma], the skew polynomial ring in t, over L, with automorphism sigma. We prove that the multiplicative group D(x) of D contains a free noncyclic subgroup.

On automorphisms of finite abelian p-groups

Let A be a finitely generated abelian group. We describe the automorphism group Aut(A) using the rank of A and its torsion part p-part A(p). For a finite abelian p-group A of type (k(1),..., k(n)), simple necessary and sufficient conditions for an n x n-matrix over integers to be associated with an automorphism of A are presented. Then, the automorphism group Aut(A) for a finite p-group A of type (k(1), k(2)) is analyzed. (C) 2008 Mathematical Institute Slovak Academy of Sciences.