Topics Covered:
Preliminaries: Basics: Sets of numbers, functions, equivalence relations, partitions. P Integers: WOP, division algorithm, Euclidean algorithm, FTArith, Euler phi-function. The integers modulo n. Groups: Definitions. Basic properties. Examples: Dihedral, symmetric, matrix, and the quaternion group. Group actions. Subroups: Definitions, examples, centralizers & normalizers, cyclic, and generated. The lattice of subgroups. Quotient groups and homomorphisms: Definitions, examples. Lagrange's theorem. The isomorphism theorems. Composition series and the Holder program. Transpositions and the Alternating group. Group actions: Permutation representations. Cayley's theorem. The class equation. Automorphisms. The Sylow theorems. The simplicity of A_n.WRITING UP HOMEWORK SOLUTIONS: You are expected to show all essential steps in solving a problem. Draw appropriate figures. If in doubt, PUT IT IN! Make your solution flow. Ask yourself the question: "If I were the reader, how would I score this problem?'' Numerical answers alone, single words "yes" or "no", etc., are NEVER, EVER acceptable as complete answers. One of the skills you are expected to learn in this class is that of communicating mathematics clearly and completely in writing. Writing up homework exercises is your opportunity to hone these skills. Your grade depends on it!
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