# 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.