Abstr Alg: Naturality & Well-Defined Mappings in Proofs of Cayley's Theorem & Orbit-Stabilizer Thm

Cayley's Theorem says that every group is isomorphic to a group of permutations. Given a group G, let G' be the set of all permutations that map x to gx, as g varies over g. This is a permutation group. The natural isomorphism from G onto G' maps a given g to this mapping (that maps each x to gx, written T_{g}, which defines a group action of G on G). The Orbit-Stabilizer Theorem says that: given a finite group of permutations G on a set S, and given any element i of S, the order of G, written |G|, is equal to the product of the order of the stabilizer of i in G |stab_{G}(i)| with the order (cardinality) of the orbit |orb_{G}(i)|. Lagrange's theorem implies that we need to prove that the index of the stabilizer of i in G equals |orb_{G}(i)|. The natural mapping is T(alpha*stab_{G}(i))=alpha(i). But we must show this is a well-defined function (well-defined mapping). This requires properties of cosets.

⏱️TIMESTAMPS⏱️
(0:00) Intro
(0:34) Cayley's Theorem statement
(2:10) Naturality in the proof of Cayley's Theorem
(3:43) The group action for a given g in G
(6:10) The group action is one-to-one
(7:49) The group action is onto
(10:51) The left regular representation G' of G
(12:29) The natural isomorphism from G to G'
(15:27) It is onto.
(17:51) It is one-to-one
(19:05) It is operation-preserving
(21:39) Lagrange's Theorem proof
(24:52) The stabilizer of a point
(28:04) The orbit of a point
(30:30) The Orbit-Stabilizer Theorem
(31:43) Proof of Orbit-Stabilizer Theorem
(34:32) The natural mapping
(39:05) The mapping is well-defined
(44:56) External direct products
(49:26) Normal subgroups and factor groups

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#cayleystheorem #orbitstabilizertheorem #groupaction

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