Abstract Algebra with Applications

• Vector Spaces First Introduction: Affine Geometry Second Introduction: Linear Equations Vector Spaces Linear and Affine Mappings Abstract Affine Geometry Representations of Linear Mappings by Matrices Determinants Volume Functions Eigenvalues and Eigenvectors Classification of Endomorphisms Up to Similarity Tensor Products and Base-Field Extension Metric Geometry Euclidean Spaces Linear Mappings Between Euclidean Spaces Bilinear Forms Groups of Automorphisms Application: Markov Chains Application: Matrix Calculus and Differential Equations Groups Introduction: Symmetries of Geometric Figures Groups Subgroups and Cosets Symmetric and Alternating Groups Group Homomorphisms Normal Subgroups and Factor Groups Free Groups: Generators and Relations Group Actions Group-Theoretical Applications of Group Actions Nilpotent and Solvable Groups Topological Methods in Group Theory Analytical Methods in Group Theory Groups in Topology Appendix Bibliography

". . .these two volumes constitute an outstanding and unique text on algebra and its various applications, which is matchlessly rich in content, comprehensive and well-arranged. . .[a] masterpiece. . . .the mathematical community should have good reason to be grateful to the author for having added a new standard text to the existing literature. . .with many distinctive and unique features, and. . .comparable to the celebrated standard texts. . ..Due to its peerless width of the material. . .this textbook. . .is certainly predestined to become an indispensable tool kit for everyone needing and using algebraic methods. "---Zentralblatt fur Mathematik