Discoveries in finite semigroups have influenced several mathematical fields, including theoretical computer science, tropical algebra via matrix theory with coefficients in semirings, and other areas of modern algebra. This comprehensive, encyclopedic text will provide the reader - from the graduate student to the researcher/practitioner with a detailed understanding of modern finite semigroup theory, focusing in particular on advanced topics on the cutting edge of research. Key features: (1) Develops q-theory, a new theory that provides a unifying approach to finite semigroup theory via quantization; (2) Contains the only contemporary exposition of the complete theory of the complexity of finite semigroups; (3) Introduces spectral theory into finite semigroup theory; (4) Develops the theory of profinite semigroups from first principles, making connections with spectra of Boolean algebras of regular languages; (5) Presents over 70 research problems, most new, and hundreds of exercises. Additional features: (1) For newcomers, an appendix on elementary finite semigroup theory; (2) Extensive bibliography and index. The q-theory of Finite Semigroups presents important techniques and results, many for the first time in book form, and thereby updates and modernizes the literature of semigroup theory.

The q-theory of Finite Semigroups

This is a survey paper on various topics concerning self-similar groups and branch groups with a focus on those notions and problems that are related to a 3-generated torsion 2 group of intermediate growth G, constructed by the author in 1980, and its generalizations Gω, ω ∈ {0, 1, 2}ℕ.

Solved and Unsolved Problems Around One Group

For any reductive group G over a global function field, we use the cohomology of G-shtukas with multiple modifications and the geometric Satake equivalence to prove the global Langlands correspondence for G in the direction \"from automorphic to Galois\". Moreover we obtain a canonical decomposition of the spaces of cuspidal automorphic forms indexed by global Langlands parameters. The proof does not rely at all on the Arthur-Selberg trace formula.

Chtoucas pour les groupes réductifs et paramétrisation de Langlands globale

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory. In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public-key cryptography so far. This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.

Non-Commutative Cryptography and Complexity of Group-Theoretic Problems

The main result of this paper is the determination of all nonassociative Moufang loops of orders *31. Combinatorial type methods are used to consider a number of cases which lead to the discovery of 13 loops of the type in question and prove that there can be no others. All of the loops found are isomorphic to all of their loop isotopes, are solvable, and satisfy both Lagrange's theorem and Sylow's main theorem. In addition to finding the loops referred to above, we prove that Moufang loops of orders p, p , p or pq (for p and q prime) must be groups. Finally, a method is found for constructing nonassociative Moufang loops as extensions of nonabelian groups by the cyclic group of order 2. L Introduction. In studying algebraic objects, it is frequently useful to have many examples at one's fingertips. In the case of Moufang loops that are not groups, the scarcity of manageable examples is one of the difficulties that we have encountered. It is the purpose of this paper to begin to remedy this situation by finding all Moufang loops of order < 31.0) There are 13 such loops-one of order 12, five of order 16, one of order 20, five of order 24, and one of order 28. The order structures, nuclei and subloops of these loops are given (Tables 3, 4 and 5). All of the loops are G-loops (i.e. they are isomorphic to all of their loop isotopes) and they are solvable. Lagrange's theorem and Sylow's main theorem hold in all of them. In terms of the M^-laws of Pflugfelder [10], some of the loops are M}-loops, some are M7-loops, and some are strictly Moufang. In the course of studying these loops, we find a general method of constructing nonassociative Moufang loops as extensions of groups (see Theorem 1). We also prove that, for p and q being primes, Moufang loops of order pq or of order p" for n < 3 are groups. Presented to the Society, July 15, 1971; received by the editors November 15, 1971. AUS (MOS) subject classifications (1970). Primary 20N05.

/pdf/moufang-loops-of-small-order-2cvdlonz4u.pdf

Moufang Loops of Small Order

Roughly speaking, a variety of groups is a class of groups in which certain relations called laws or rules or identities are universally valid. The most widely investigated variety of groups is that of abelian groups in which the commutative law is universally valid or, in other words, in which for arbitrary elements x1, x2 of the group, the identity 

$$ \left( {{x_1},{x_2}} \right) = 1 $$

holds, where, as usual, (x1, x2) is defined as the commutator 

$$ x_1^{ - 1}x_2^{ - 1}{x_1}{x_2} $$

of x1 and x2. Similarly, the Burnside variety of exponent e is defined as the class of groups in which, for all elements x, the relation 

$$ {x^e} = 1 $$

holds. The metabelian groups of Chapter II.6 can be defined as the variety of groups define4 by the identity 

$$ \left( {\left( {{x_1},{x_2}} \right),\left( {{x_3},{x_4}} \right)} \right) = 1, $$

and the nilpotent groups of class c discussed in Chapter II.7 form the variety defined by the identity 

$$ \left( {{x_1},{x_2},...,{x_c},{x_{c + 1}}} \right) = 1 $$

where the parentheses ( ) denote the simple (c + l)-fold commutator defined in Chapter II.7.

Varieties of Groups

I The Beginning of Combinatorial Group Theory.- I.I Introduction to Part I.- I.2 The Foundations: Dyck's Group-Theoretical Studies.- I.3 The Origin: The Theory of Discontinuous Groups.- I.4 Motivation: The Fundamental Groups of Topological Spaces.- I.5 The Graphical Representation of Groups.- I.6 Precursors of Later Developments.- A. Arithmetically Defined Linear Groups in Higher Dimensions.- B. Arithmetically Defined Linear Groups in Two Dimensions.- C. Geometric Constructions. Fuchsian Groups.- D. Braid Groups and Mapping Class Groups.- E. Differential Equations, Linear Groups, and Lie Groups.- F. Finite Groups.- I.7 Summary.- I.8 Modes of Communication. Growth and Distribution of Research in Group Theory.- I.9 Biographical Notes.- I.10 Notes on Terminology and Definitions.- I.11 Sources.- II The Emergence of Combinatorial Group Theory as an Independent Field.- II. 1 Introduction to Part II.- II.2 Free Groups and Their Automorphisms.- II.3 The Reidemeister-Schreier Method.- II.4 Free Products and Free Products with Amalgamations.- II.5 One-Relator Groups.- II.6 Metabelian Groups and Related Topics.- A. The Principal Ideal Theorem.- B. Applications to the Theory of Knots and Links.- C. A Problem from the Foundations of Geometry.- D. Notes on Later Developments and Generalizations.- II.7 Commutator Calculus and the Lower Central Series.- II.8 Varieties of Groups.- II.9 Topological Properties of Groups and Group Extensions.- II.10 Notes on Special Groups.- II.11 Postscript: The Impact of Mathematical Logic.- II.12 Modes of Communication.- II.13 Geographical Distribution of Research and Effects of Migration.- II.14 Organization of Knowledge.- Index of Names.- Index of Subjects.

The History of Combinatorial Group Theory : A Case Study in the History of Ideas

Theory and Problems of Group Theory.

There exist two phenomena in the development of mathematics in the Twentieth Century which have no precedents in earlier times. They are the rates of increase in the volume of knowledge and in the complexity of proofs. We shall start with the first of these by reproducing below a statistical table which is due to W. R. Utz of the University of Missouri and was published in the Notices of the American Mathematical Society, Issue 211, p. 424, August 1981. We have already mentioned this table in Chapter 11.12 because of its two right-hand columns. What concerns us here are the two left-hand columns which show the number of titles of mathematical papers which appeared in the Mathematical Reviews during the last years of five consecutive decades.

Organization of Knowledge

Sets, Mappings and Binary Operations, Groupoids.Groups and Subgroups.Isomorphism Theorems.Finite Groups.Abelian Groups.Permutational Representations.Free Groups and Presentations.Appendices: A: Number Theory.B: Guide to the Literature.Symbols and Notations.

Schaum's outline of theory and problems of group theory

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