Abstract Algebra, Lec 10B: Symmetric Group S3, Generators & Relations, Permutation Properties

"Contemporary Abstract Algebra", by Joe Gallian: https://amzn.to/2ZqLc1J. Amazon Prime Student 6-Month Trial: https://amzn.to/3iUKwdP. 🔴 Check out my blog at: https://infinityisreallybig.com/ 🔴 Abstract Algebra Playlist: https://www.youtube.com/watch?v=lx3qJ-zjn5Y&list=PLmU0FIlJY-Mn3Pt-r5zQ_-Ar8mAnBZTf2 🔴 Abstract Algebra Problems with Solutions (including Proofs): https://www.youtube.com/playlist?list=PLmU0FIlJY-MlqikmY6khGUZRueXwsRFQV (0:00) Check the claim that S3 can be represented as combinations of a 3-cycle and a 2-cycle (transposition). Use array notation to start with. (4:46) Cycle notation and composition ("products") of permutations in cycle notation. (10:32) Generators and relations for S3 and their use to do group computations and create a Cayley table. (17:44) S3 is isomorphic to D3 (note that alpha and beta could have been chosen differently). (20:30) Every permutation of a finite set can be written as a cycle or as a product of disjoint cycles. (22:45) Disjoint cycles commute, but non-disjoint cycles typically do not commute. (24:31) Order of a permutation of a finite set written in disjoint cycle form (it's the lcm of the lengths of the cycles) and note the fact that Sn is not a subgroup of Sm when n and m are different, though in cycle notation we are free to "think of" a given element as being in some Sn or some Sm of higher degree. (26:58) Use Mathematica to confirm the order of the example is 12 (it's a composition of disjoint cycles, one of length 3 and the other of length 4, so lcm(3,4) = 12). (29:16) Permutations as products of two cycles, even & odd permutations, alternating group An of order n!/2. AMAZON ASSOCIATE As an Amazon Associate I earn from qualifying purchases.