By Israel Kleiner

ISBN-10: 0817646841

ISBN-13: 9780817646844

This e-book does not anything below offer an account of the highbrow lineage of summary algebra. the advance of summary algebra used to be propelled by means of the necessity for brand spanking new instruments to deal with definite classical difficulties that seemed insoluble via classical skill. an immense subject matter of the e-book is to teach how summary algebra has arisen in trying to resolve a few of these classical difficulties, offering a context from which the reader could achieve a deeper appreciation of the math concerned. arithmetic teachers, algebraists, and historians of technology will locate the paintings a invaluable reference.

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The crowning achievement of these two lines of development—a symphony on the grand themes of Galois and Cauchy—was Jordan’s important and influential Treatise on Substitutions and Algebraic Equations of 1870. Although the author stated in the preface that “the aim of the work is to develop Galois’ method and to make it a proper field of study, by showing with what facility it can solve all principal problems of the theory of equations,” it is in fact group theory per se—not as an offshoot of the theory of solvability of equations—which formed the central object of study.

4 Consolidation of the abstract group concept; dawn of abstract group theory 33 Von Dyck’s definition of an abstract group, which included both the finite and infinite cases, was given in terms of generators (he calls them “operations”) and defining relations (the definition is somewhat long—see [7]). He stressed that “in this way all . . ” He then went on to construct the free group on n generators, and showed (essentially, without using the terminology) that every finitely generated group is a quotient group of a free group of finite rank.

Klein’s use of groups in geometry was, of course, explicit. ) In the next section we will note the significance of Klein’s Erlangen Program (and his other works) for the evolution of group theory. Since the Program originated a hundred years after Lagrange’s work and eighty years after Gauss’ work, its importance for group theory can best be appreciated after a discussion of the evolution of group theory beginning with the works of Lagrange and Gauss and ending with the period around 1870. 4 Analysis In 1874 Lie introduced his general theory of continuous transformation groups— essentially what we call Lie groups today.

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A History of Abstract Algebra by Israel Kleiner


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