Abstract Algebra, Lec 19B: Factor Group Applications (Cauchy's Theorem), Internal Direct Products скачать в хорошем качестве

Abstract Algebra, Lec 19B: Factor Group Applications (Cauchy's Theorem), Internal Direct Products

"Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J Check out my blog at: https://infinityisreallybig.com/ Abstract Algebra Playlist:    • Abstract Algebra Course, Lecture 1: Introd...   (0:00) Make sure you truly understand what it means for (left) coset multiplication to be well-defined. (0:58) Outline the rest of the proof that the factor (quotient) group G/H is well-defined. (4:05) The G/Z Theorem and corollaries (including example where G = D6: it turns out that D6/Z(D6) is isomorphic to D3 (and S3) and the fact that if G is a non-Abelian group of order pq, then Z(G) = {e}). (11:08) G/Z(G) is isomorphic to Inn(G). (12:32) Cauchy's Theorem for Abelian groups (a partial converse of Lagrange's Theorem) (a special case of the First Sylow Theorem). (14:24) The proof uses factor groups and induction (one of the most powerful proof techniques in abstract algebra because G/H will be "smaller" than G if H is not {e}). (15:55) Internal direct product (of two subgroups) definition and unique representation consequence (and discuss analog from linear algebra). (21:22) If G = H x K, then, G is isomorphic to H + K and discuss how the isomorphism would be defined. (23:55) General philosophy of external versus internal direct products. (25:05) Another classification fact and corollary (groups of order p^2 are either cyclic or isomorphic to the external direct product of cyclic groups of order p, and are therefore Abelian). (27:19) Notes about proofs to study. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.