A group action (or G-set) is transitive if its action graph is connected. A natural question to ask is “how can we classify all connected action graphs?” The action of G on the cosets of a subgroup H is an example, which is constructed by collapsing the Cayley graph by the right cosets of H. It turns out every connected action graph arises in this manner. Said differently, every transitive G-set is isomorphic to the quotient of G by one of its subgroups. Yes, we can take the quotient of G by any subgroup H, normal or not, though the resulting G-set is a group if and only if H is normal. This leads to a Galois correspondence between conjugacy classes of subgroups, and G-sets, or action graphs, and we’ll see some beautiful visuals that illustrate this. We’ll conclude with a few examples of theorems involving subgroups of a certain index being normal, where the proof begins as “Let G act on the cosets of H by multiplication…,” and we’ll discuss what that really means.
Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule/visualalgebra/
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0:00 Introduction
1:31 Motivating question: how to classify the action graphs of a group?
3:23 Collapsing left cosets vs. right cosets
8:23 The transitive D₄-sets
10:10 Every transitive action is isomorphic to G acting on cosets
18:12 Three D₆-set of order 6.
22:31 Conjugates of stab(s) give the same G-set (visual)
25:09 Conjugates of stab(s) give the same G-set (proof)
32:24 Big idea summary of transitive G-sets
33:24 The transitive D₆-sets and a Galois correspondence
35:26 A sufficient condition for index-3 subgroups being normal
43:43 A sufficient condition for index-p subgroups being normal