Visual Algebra, Lecture 5.9: Equivariance and G-set homomorphisms

Описание: Previously, we’ve seen that a G-set isomorphism is a bijection that commutes with the action. In this lecture, we’ll start with the concept of a G-set automorphism. These form a group, and they can be thought of as symmetries of the action graph. If H is the stabilizer of any element, then the G-set automorphism group is the quotient of the normalizer of H, by H itself. This arises in algebraic topology as the group of deck transformations of a covering space. We’ll also look at the concept of a G-set homomorphism. Finally, we’ll interpret the orbit-stabilizer theorem as the fundamental homomorphism theorem for G-sets. Namely, that every transitive G-set is just the quotient of G by the cosets of some subgroup.

Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule...

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0:00 Introduction
1:08 Action equivalence vs. G-set isomorphism
2:47 Equivariant maps
5:49 A D₆-set automorphism
8:30 Automorphism of three different D₆-sets
10:31 Two G-set automorphism groups of size 4
12:44 Action graphs with C₄- and V₄-symmetries
13:25 Theorem and proof outline: Aut(H/G) ≅ N(H)/H
16:18 Lemma 1: Every G-set automorphism has the form Hg↦Hxg
18:30 Lemma 2: If Hg↦Hxg is a G-set automorphism, then x∈N(H)
21:55 Lemma 2: If x∈N(H), then Hg↦Hxg is a G-set automorphism
25:39 Proof of Aut(H/G) ≅ N(H)/H
30:56 N(H) vs. Ker(ϕ) in the subgroup lattice
34:07 G-set homomorphisms
36:11 An example of a D₆-set homomorphism
36:33 The "fundamental homomorphism theorem for G-sets"