### Linear Algebraic Groups (Part 4)

#### Posted by John Baez

This time I explain some axioms for an ‘abstract projective plane’, and the extra axiom required to ensure an abstract projective plane comes from a field. Yet again the old Greek mathematicians seem to have been strangely prescient, because this extra axiom was discovered by Pappus of Alexandria sometime around 340 AD!
For him it was a theorem in Euclidean geometry, but later it was realized that a cleaner statement involves only projective geometry… and later still, it was seen to be a useful *axiom*.

For details, read the notes.

- Lecture 4 (Oct. 4) - Abstract projective planes. Pappus’s hexagon theorem, and how it characterizes which abstract projective planes are of the form $k\mathrm{P}^2$ for a field $k$. Klein geometry and transitive group actions: each kind of highly symmetrical geometry corresponds to a group $G$, and each type of geometrical figure in this geometry corresponds to a set on which $G$ acts transitively. Transitive $G$-spaces all arise from subgroups of $G$. Klein geometry studies invariant relations between transitive $G$-spaces.

The subject of abstract projective planes touches on the fascinating axiomatic approach to incidence geometries having various linear algebraic groups as their symmetry groups. But instead of marching down that beckoning byway, I’ll point you to a place where you can read more:

- Eric Moorhouse,
*Incidence Geometry*.

I want to head down the main road: Klein’s Erlangen program!

## Re: Linear Algebraic Groups (Part 4)

Thanks for these notes: all very fascinating!

I’ve found what I believe are some typos, although it’s always possible that I’ve misunderstood things; if so, please forgive me.

Lecture 2, page 1: in the definition of the sphere X, the dot product v.c should read “v.v”

Lecture 2, page 2: in the definition of the set of points P, I believe that instead of reading “P={p : p is a 1-dim subspaces of k

^{3}} n X” this should read “{P={p n X : p is a 1-dim subspaces of k^{3}}”. And same for L. (Using “n” for set intersection; also, “subspaces” should be singular.)Lecture 3, page 1: the definition of “kP

^{n}” should instead say kP^{n-1}(at least, that’s how it was at the end of Lecture 2, which I believe is correct.)