Abstract Alg, Lec 15B: More on Cosets, Lagrange's Theorem, Converse is False, Orbit-Stabilizer Thm

(0:00) Encouragement to work hard on knowing coset properties and their proofs.
(0:26) More properties of cosets.
(2:18) Visualizing the idea of the proof of Lagrange's theorem.
(6:16) This partition is generated by a special equivalence relation: a equiv b iff aH = bH iff a^(-1)b is in H (which generates equivalence classes with the same number of elements).
(7:43) The index of a subgroup H in a group G (and an example where G is an infinite group) and its relationship to the proof of Lagrange's Theorem.
(11:06) The converse of Lagrange's Theorem is false (Example: A4, which has order 12, has no subgroup of order 6).
(14:10) Stabilizers of elements in a set S under a permutation group.
(16:08) Orbits of elements in a set S under a permutation group.
(17:48) Statement of the Orbit-Stabilizer Theorem and mention applications.
(19:48) The idea of the proof of the Orbit-Stabilizer Theorem (define a one-to-one and onto mapping that must first be shown to be well-defined).

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