Titlebook: Algebra; An Approach via Modu William A. Adkins,Steven H. Weintraub Textbook 1992 Springer Science+Business Media New York 1992 Permutatio

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Groups,In this chapter we introduce groups and prove some of the basic theorems in group theory. One of these, the structure theorem for finitely generated abelian groups, we do not prove here but instead derive it as a corollary of the more general structure theorem for finitely generated modules over a PID (see Theorem 3.7.22).

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Rings,(1.1) Definition. . ring (.,+,) . +: . ×.→. (.) . : . ×.→. (.) ..

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Group Representations,We begin by defining the objects that we are interested in studying. Recall that if . is a ring and . is a group, then .(.) denotes the group ring of . with coefficients from .. The multiplication on .(.) is the convolution product (see Example 2.1.10 (15)).

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Algebra978-1-4612-0923-2Series ISSN 0072-5285 Series E-ISSN 2197-5612

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Linear Algebra,al form theory for a linear transformation from a vector space to itself. The fundamental results will be presented in Section 4.4. We will start with a rather detailed introduction to the elementary aspects of matrix algebra, including the theory of determinants and matrix representation of linear

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Matrices over PIDs,y if the .[.]-modules . and . are isomorphic (Theorem 4.4.2). Since the structure theorem for finitely generated torsion .[.]-modules gives a criterion for isomorphism in terms of the invariant factors (or elementary divisors), one has a powerful tool for studying linear transformations, up to simil

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Bilinear and Quadratic Forms, means of the operations (.+.)(.)=.(.)+.(.) and (.)(.)= .(.(.)) for all .. Moreover, if . then Hom.(.)= End .(.) is a ring under the multiplication (.)(.)=.(.(.)). An .-module ., which is also a ring, is called an .-algebra if it satisfies the extra axiom .(.)=(.).=.(.) for all . ∈ . and . ∈ .. Thus