Group Theory with Power Tools

Note

This is just a quick sketch of the class, and will be filled in likely only if the class it taught again.

What is a Group

Laws of composition. Identity and inverse.

Working examples of groups:

  • Cutting group: cut at position \(x\), look at cuts. inverse is screws.
  • Dihedral group we'll manufacture.
  • Lattices & finite abelian groups.

Subgroups

Ignore part of the figure to get a subgroup. Or, link some points together. Or restrict the points at which you may cut.

Isomorphism and Homomorphism

Make two things with the same dihedral group. Look at permutations of numbers as something to homomorphism with.

Symmetry

Groups can represent symmetry. Conjugation. Relation to commutativity. Normal subgroups. Let's make some platonic solids.

Quotients

We ignore some distinguishing feature of two configurations, making some element the identity.

Lattices

A lattice is a finitely generated abelian group under vector addition.

Classification Theorem of Finitely Generated Abelian Groups

Make each generator a unit vector in a new orthogonal dimension. Then we can add each generator until it wraps around, or doesn't, to make a full lattice. Now consider each dimension indiviually, giving you cyclic group in each dimension. This shows that any finitely generated abelian group is the product of cyclic groups.

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