Abstract Alg, Lec 16B: Proof of Orbit-Stabilizer Theorem, External Direct Product Examples and Facts

(0:00) Reminder that coset properties are very important.
(0:30) Outline of the proof of the Orbit-Stabilizer Theorem (with an emphasis on proving that a one-to-one and onto mapping (bijection) is well-defined).
(10:50) Definition of the external direct product of a finite number of groups (as a set it's a Cartesian product and the operation is done component-wise (component by component)) and the notation for it in our text (by Gallian).
(15:55) Example 1: the external direct product of Z2 with itself and its Cayley table.
(19:54) Example 2: the external direct product of Z2 and Z3 (as a set and a couple computations).
(25:18) Example 3: the external direct product of Z2 and D3 (group of symmetries of an equilateral triangle).
(27:56) Important results about the order of an external direct product of finitely many finite groups and the order of an element in an external direct product is the lcm of the orders of the components of the element.

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