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

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(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.

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Видео Abstract Algebra, Lec 10B: Symmetric Group S3, Generators & Relations, Permutation Properties канала Bill Kinney