In the previous lecture, we saw the first of the four isomorphism theorems, the “Fundamental Homomorphism Theorem”. It says that every homomorphic image is a quotient. The next natural question to ask, is “what can we say about the structure of a quotient?” In particular, what are its subgroups? Which ones are normal? What are the conjugacy classes? These questions and more are addressed by the next isomorphism theorem, the correspondence theorem, which is the focus of this lecture. We’ll prove the main part, and explore it through a visual analogy that I call “shoebox diagrams,” which I think is unique to my book and video series. We will also heavily rely on subgroup lattices throughout.
Course & book webpage (with complete lecture note slides, HW, exams, etc.): https://www.math.clemson.edu/~macaule/visualalgebra/
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CHAPTERS
0:00 Introduction
0:56 The isomorphism theorems
5:05 The correspondence theorem, informally
6:43 Shoebox diagrams and Cayley graphs
10:30 Shoebox diagrams and subgroup lattices
14:43 The correspondence theorem formally
17:28 A subgroup lattice interpretation of the correspondence theorem
20:50 Conjugacy classes and the correspondence theorem
26:29 Proof of the correspondence theorem
33:33 G/Z(G) can never be a nontrivial cyclic group