By the fundamental homomorphism theorem, we know that every homomorphic image is a quotient. By the correspondence theorem, we know how to characterize the subgroups of a quotient, and which are normal. The next question to ask is how can we characterize the quotients by these normal subgroups. In other words, what can we say about a quotient of a quotient group? This is answered by the 3rd isomorphism theorem, that I call the “fraction theorem.” It has very aesthetically pleasing interpretations both in terms of subgroup lattices, and of Cayley graphs with nested “shoeboxes” --- cosets. You won’t see these anywhere else! Once we gain intuition of this theory, we’ll prove it, which is surprisingly straightforward.
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
2:36 Cayley graph view of ϕ=ι∘π decomposition
3:40 Subgroup lattice view of ϕ=ι∘π decomposition
4:30 The quotient of a subgroup in SL₂(ℤ₃)
6:02 The subgroup of a quotient in SL₂(ℤ₃)
9:39 The fraction theorem
12:29 Lattice interpretation of the fraction theorem
13:31 The fraction theorem with Cayley graphs and shoeboxes
15:51 Proof of the fraction theorem
23:13 Another shoebox picture of the fraction theorem