This book gives an elementary treatment of the basic material about graph spectra, both for ordinary, and Laplace and Seidel spectra. The text progresses systematically, by covering standard topics before presenting some new material on trees, strongly regular graphs, two-graphs, association schemes, p-ranks of configurations and similar topics. Exercises at the end of each chapter provide practice and vary from easy yet interesting applications of the treated theory, to little excursions into related topics. Tables, references at the end of the book, an author and subject index enrich the text. Spectra of Graphs is written for researchers, teachers and graduate students interested in graph spectra. The reader is assumed to be familiar with basic linear algebra and eigenvalues, although some more advanced topics in linear algebra, like the Perron-Frobenius theorem and eigenvalue interlacing are included.

Spectra of graphs

This paper introduces five notions, including π-algebras, π-ideals, Hopf π-algebras, π-modules and Hopf π-modules, verifies the fundamental isomorphism theorem of π-algebras and studies some algebraic properties of Hopf π-algebras as well.

Hopf π- Algebras

General introduction to the special odd case General lemmas Theorem $C^*_2$: Stage 1 Theorem $C^*_2$: Stage 2 Theorem $C_2$: Stage 3 Theorem $C_2$: Stage 4 Theorem $C_2$: Stage 5 Theorem $C_3$: Stage 1 Theorem $C_3$: Stages 2 and 3 IV$_K$: Preliminary properties of $K$-groups Background references Expository references Glossary Index.

http://inis.jinr.ru/sl/M_Mathematics/MA_Algebra/MAtg_Group%20theory/Gorenstein%20D.,%20Lyons%20R.,%20Solomon%20R.%20Classification%20of%20finite%20simple%20groups%201%20(AMS%20survey%2040%20no.1,%201994,%202000)(176s).pdf

The Classification of the Finite Simple Groups

This book is designed as a text for the first year of graduate algebra, but it can also serve as a reference since it contains more advanced topics as well. This second edition has a different organization than the first. It begins with a discussion of the cubic and quartic equations, which leads into permutations, group theory, and Galois theory (for finite extensions; infinite Galois theory is discussed later in the book). The study of groups continues with finite abelian groups (finitely generated groups are discussed later, in the context of module theory), Sylow theorems, simplicity of projective unimodular groups, free groups and presentations, and the Nielsen-Schreier theorem (subgroups of free groups are free). The study of commutative rings continues with prime and maximal ideals, unique factorization, noetherian rings, Zorn's lemma and applications, varieties, and Grobner bases. Next, noncommutative rings and modules are discussed, treating tensor product, projective, injective, and flat modules, categories, functors, and natural transformations, categorical constructions (including direct and inverse limits), and adjoint functors. Then follow group representations: Wedderburn-Artin theorems, character theory, theorems of Burnside and Frobenius, division rings, Brauer groups, and abelian categories. Advanced linear algebra treats canonical forms for matrices and the structure of modules over PIDs, followed by multilinear algebra. Homology is introduced, first for simplicial complexes, then as derived functors, with applications to Ext, Tor, and cohomology of groups, crossed products, and an introduction to algebraic $K$-theory. Finally, the author treats localization, Dedekind rings and algebraic number theory, and homological dimensions. The book ends with the proof that regular local rings have unique factorization. (GSM/114)

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Advanced Modern Algebra

This paper reports on a six-year collaborative effort that culminated in a complete formalization of a proof of the Feit-Thompson Odd Order Theorem in the Coq proof assistant. The formalized proof is constructive, and relies on nothing but the axioms and rules of the foundational framework implemented by Coq. To support the formalization, we developed a comprehensive set of reusable libraries of formalized mathematics, including results in finite group theory, linear algebra, Galois theory, and the theories of the real and complex algebraic numbers.

/pdf/a-machine-checked-proof-of-the-odd-order-theorem-4dpiu9cfrx.pdf

A machine-checked proof of the odd order theorem

Algebras, modules, and representations Group representations and characters Characters and integrality Products of characters Induced characters Normal subgroups T.I. sets and exceptional characters Brauer's theorem Changing the field The Schur index Projective representations Character degrees Character correspondence Linear groups Changing the characteristic Some character tables Bibliographic notes References Index.

Character theory of finite groups

Sylow theory Subnormality Split extensions Commutators Transfer Frobenius actions The Thompson subgroup Permutation groups More on subnormality More transfer theory The basics Index.

Finite Group Theory

This book, based on a first-year graduate course the author taught at the University of Wisconsin, contains more than enough material for a two-semester graduate-level abstract algebra course, including groups, rings and modules, fields and Galois theory, an introduction to algebraic number theory, and the rudiments of algebraic geometry. In addition, there are some more specialized topics not usually covered in such a course. These include transfer and character theory of finite groups, modules over artinian rings, modules over Dedekind domains, and transcendental field extensions. This book could be used for self study as well as for a course text, and so full details of almost all proofs are included, with nothing being relegated to the chapter-end problems. There are, however, hundreds of problems, many being far from trivial. The book attempts to capture some of the informality of the classroom, as well as the excitement the author felt when taking the corresponding course as a student.

Algebra, a graduate course

http://www.math.wm.edu/~vinroot/SuperCharAIM.pdf

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

A finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements. (Equivalently, the degree of this character is the square root of the index of the center.) In 1964, Iwahori and Matsumoto [15] conjectured that a group of central type must be solvable. The paper of Liebler and Yellen [16] aims to prove this, but there is a gap in their proof (as explained below). Nevertheless, they do correctly obtain substantial information about a minimal counterexample to the conjecture. The aim of this paper is to fill the gap in the Liebler-Yellen proof (using as they do, the classification of finite simple groups) and to provide further information about which solvable groups can have central type. We show, for instance, that every normal subgroup of the central factor group of a (solvable) group of central type has the property that its system normalizers have square index. It is interesting to contrast this restrictive condition with the result of Gagola that every solvable group is embeddable in the central factor group of a group of central type. (See [6, Theorem 1.2]). The arguments in [16] show that a minimal counterexample G to the Iwahori -Matsumoto conjecture must have the following structure: If K/Z is any minimal normal subgroup of G/Z (where Z = Z ( G ) , the center), then K is abelian, K/Z is a 2-group, S = CG(K ) is the maximal solvable normal subgroup of G and any minimal normal subgroup HIS of G/S is the direct product of a number l of copies of GL(3, 2). Indeed, the Liebler-Yellen methods can be used to show that the Sylow 2-subgroup of K, considered as an (H/S)-module is a product o f / copies of a GL(3,2)-module of dimension 4, this module having a unique minimal submodule, which is trivial, and an irreducible factor module of dimension 3. The minimal submodules are amalgamated in the product.

I. Martin Isaacs

Papers

Character theory of finite groups

Finite Group Theory

Algebra, a graduate course

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

On groups of central type