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This is the first book to contain a rigorous construction and uniqueness proof for the largest and most famous sporadic simple group, the Monster. The author provides a systematic exposition of the theory of the Monster group, which remains largely unpublished despite great interest from both mathematicians and physicists due to its intrinsic connection with various areas in mathematics, including reflection groups, modular forms and conformal field theory. Through construction via the Monster amalgam - one of the most promising in the modern theory of finite groups - the author observes some important properties of the action of the Monster on its minimal module, which are axiomatized under the name of Majorana involutions. Development of the theory of the groups generated by Majorana involutions leads the author to the conjecture that Monster is the largest group generated by the Majorana involutions.

Involutes (Mathematics) --- Sporadic groups (Mathematics) --- Involutes (Mathematics). --- Sporadic groups (Mathematics). --- Curves --- Inversions (Geometry) --- Groups, Sporadic (Mathematics) --- Finite groups

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A Course on Finite Groups introduces the fundamentals of group theory to advanced undergraduate and beginning graduate students. Based on a series of lecture courses developed by the author over many years, the book starts with the basic definitions and examples and develops the theory to the point where a number of classic theorems can be proved. The topics covered include: group constructions; homomorphisms and isomorphisms; actions; Sylow theory; products and Abelian groups; series; nilpotent and soluble groups; and an introduction to the classification of the finite simple groups. A number of groups are described in detail and the reader is encouraged to work with one of the many computer algebra packages available to construct and experience "actual" groups for themselves in order to develop a deeper understanding of the theory and the significance of the theorems. Numerous problems, of varying levels of difficulty, help to test understanding. A brief resumé of the basic set theory and number theory required for the text is provided in an appendix, and a wealth of extra resources is available online at www.springer.com, including: hints and/or full solutions to all of the exercises; extension material for many of the chapters, covering more challenging topics and results for further study; and two additional chapters providing an introduction to group representation theory.

Electronic books. -- local. --- Finite groups. --- Mathematics. --- Finite groups --- Mathematics --- Algebra --- Physical Sciences & Mathematics --- Math --- Groups, Finite --- Algebra. --- Group theory. --- Science education. --- Science Education. --- Group Theory and Generalizations. --- Science --- Science education --- Scientific education --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematical analysis --- Group theory --- Modules (Algebra) --- Study and teaching.

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This book contributes to important questions in the representation theory of finite groups over fields of positive characteristic — an area of research initiated by Richard Brauer sixty years ago with the introduction of the blocks of characters. On the one hand, it introduces and develops the abstract setting of the Frobenius categories — also called the Saturated fusion systems in the literature — created by the author fifteen years ago for a better understanding of what was loosely called the local theory of a finite group around a prime number p or, later, around a Brauer block, and for the purpose of an eventual classification — a reasonable concept of simple Frobenius category arises. On the other hand, the book develops this abstract setting in parallel with its application to the Brauer blocks, giving the detailed translation of any abstract concept in the particular context of the blocks. One of the new features in this direction is a framework for a deeper understanding of one of the central open problems in modular representation theory, known as Alperin’s Weight Conjecture (AWC). Actually, this new framework suggests a more general form of AWC, and a significant result of the book is a reduction theorem of this form of AWC to quasi-simple groups. Although this book is a research monograph, all the arguments are widely developed to make it accessible to the interested graduate students and, at the same time, to put them on the verge of the research on this new subject: the third part of the book on the localities associated to a Frobenius category gives some insight on the open question about the existence and the uniquenes of a perfect locality — also called centric linking system in the literature. We have developed a long introduction to explain our purpose and to provide a guideline for the reader throughout the twenty four sections. A systematic appendix on the cohomology of categories completes the book.

Brauer groups. --- Finite groups. --- Frobenius algebras. --- Frobenius groups. --- Grothendieck groups. --- Group theory. --- Representations of groups. --- Frobenius algebras --- Frobenius groups --- Grothendieck groups --- Brauer groups --- Finite groups --- Representations of groups --- Mathematics --- Physical Sciences & Mathematics --- Algebra --- Group representation (Mathematics) --- Groups, Representation theory of --- Groups, Finite --- Brauer group --- Groups, Brauer --- Algebras, Frobenius --- Mathematics. --- Algebraic topology. --- Group Theory and Generalizations. --- Algebraic Topology. --- Topology --- Groups, Theory of --- Substitutions (Mathematics) --- Math --- Science

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There are so many chemical sensor books on the market that adding one more would seem to be a rather unnecessary exercise. My aim has been to provide material that college or graduate level instructors could use when teaching this subject. The focus is on operating principles, rather than on description of specific sensor or even types of sensors. People who regularly referee sensor manuscripts and research proposals may also find it useful as a reference, for testing one or another dubious proposed sensing scheme. Unfortunately, some strange ideas often get through the review process and become a permanent part of the knowledge database. A "food-for-thought" section has been added at the end of each chapter, which is intended to provoke probing questions about such "schemes". Since this is a never-ending process a "food-for-thought" website has been established as a permanent "blog" that supplements this book. It can be accessed at: http://www.chemistry.gatech.edu/faculty/Janata/.

Analytical biochemistry. --- Chemistry. --- Materials. --- Physics. --- Chemical detectors --- Semigroups --- Finite groups --- Chemical & Materials Engineering --- Chemistry --- Engineering & Applied Sciences --- Physical Sciences & Mathematics --- Chemical Engineering --- Analytical Chemistry --- Semigroups. --- Finite groups. --- Groups, Finite --- Analytical chemistry. --- Electrochemistry. --- Materials science. --- Analytical Chemistry. --- Physics, general. --- Materials Science, general. --- Group theory --- Modules (Algebra) --- Engineering --- Engineering materials --- Industrial materials --- Engineering design --- Manufacturing processes --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Analytic biochemistry --- Biochemistry --- Chemistry, Analytic --- Materials --- Bioanalytic chemistry --- Bioanalytical chemistry --- Analytical chemistry --- Material science --- Chemistry, Physical and theoretical --- Analysis, Chemical --- Analytic chemistry --- Chemical analysis

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Regular rings were originally introduced by John von Neumann to clarify aspects of operator algebras ([33], [34], [9]). A continuous geometry is an indecomposable, continuous, complemented modular lattice that is not ?nite-dimensional ([8, page 155], [32, page V]). Von Neumann proved ([32, Theorem 14. 1, page 208], [8, page 162]): Every continuous geometry is isomorphic to the lattice of right ideals of some regular ring. The book of K. R. Goodearl ([14]) gives an extensive account of various types of regular rings and there exist several papers studying modules over regular rings ([27], [31], [15]). In abelian group theory the interest lay in determining those groups whose endomorphism rings were regular or had related properties ([11, Section 112], [29], [30], [12], [13], [24]). An interesting feature was introduced by Brown and McCoy ([4]) who showed that every ring contains a unique largest ideal, all of whose elements are regular elements of the ring. In all these studies it was clear that regularity was intimately related to direct sum decompositions. Ware and Zelmanowitz ([35], [37]) de?ned regularity in modules and studied the structure of regular modules. Nicholson ([26]) generalized the notion and theory of regular modules. In this purely algebraic monograph we study a generalization of regularity to the homomorphism group of two modules which was introduced by the ?rst author ([19]). Little background is needed and the text is accessible to students with an exposure to standard modern algebra. In the following, Risaringwith1,and A, M are right unital R-modules.

Endomorphism rings. --- Homomorphisms (Mathematics). --- Modules (Algebra). --- Rings (Algebra). --- Modules (Algebra) --- Homomorphisms (Mathematics) --- Rings (Algebra) --- Algebra --- Mathematics --- Physical Sciences & Mathematics --- Algebraic rings --- Ring theory --- Finite number systems --- Modular systems (Algebra) --- Mathematics. --- Algebra. --- Associative rings. --- Group theory. --- Associative Rings and Algebras. --- Group Theory and Generalizations. --- Algebraic fields --- Functions --- Finite groups --- Groups, Theory of --- Substitutions (Mathematics) --- Mathematical analysis