When a group arises as a subgroup of another, its subgroup lattice appears at the bottom. When a group arises as a quotient, its subgroup lattice appears at the top. In this lecture, we’ll revisit the concept of a subquotient, which arises when a familiar subgroup lattice appears in the middle of a bigger lattice. Our last isomorphism theorem, the diamond theorem, describes a certain duality that subquotients have a subgroup lattice, that it easy to spot and visually pleasing. We will see subquotients appear in a few other setting, such as the Butterfly lemma of Hans Zassenhaus, and involving subgroups called commutators. I will also show you an original visual of the diamond theorem that Pulitzer Prize winning author Doug Hofstadter shared with me, that he calls a "pizza diagram."
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
1:05 Summary of the isomorphism theorems
2:24 A subquotient in SL₂(ℤ₃)
3:22 Statement of the diamond theorem
5:10 Proof that A∩B ⊴ A and B ⊴ AB
8:38 Proof that A/(A∩ B) ≅ AB/B
14:23 Corollary: |AB| = |A|⋅|B| / |A∩B|
16:53 Subquotient duality in the ℤ₂×ℤ₆ subgroup lattice
17:42 Subquotient duality in the Q₈⋊C₉ subgroup lattice
19:17 Pizza diagrams and "Gödel, Escher, Bach"
23:13 An application to permutation groups
26:47 The butterfly lemma
29:01 Commutators
34:15 The abelianization of a group
38:35 Commutators subgroups & abelianizations in subgroup lattices
42:08 Higher order commutator subgroups