dcernst
commented
3 years ago One possible approach (not necessarily in the following order):
- Shortly after doing isomorphisms, ask what happens when we remove surjective (embedding) or injective.
- Introduce kernel
- Introduce clones and cosets
- Build up to First Isomorphism Theorem (need aker(phi) = bker(phi) iff phi(a) = phi(b)).
- Build up to Lagrange's Theorem (should this come before or after FIT). My current approach is too cumbersome: I use both partition and equivalence relation approach; both aren't needed.
- Motivate normal subgroups and quotient groups after dealing with case of using kernel.
- Matt has a nice discussion involving taking quotients of products.
- I should introduce things like Z2xZ3, Z3xZ3, Z2xZ4 early on (using actions on "slot machines" perhaps).
- Maybe introduce rigid symmetries of tetrahedron and cube early on as examples.
- Matt and VGT have nice visuals of products involving tables. I should also do more examples of tables for quotients.
I need to another round of a big picture overhaul to cover material quicker. I'd love to get Lagrange sooner and I'd also like to get homomorphisms sooner. But I can do everything sooner.