User talk:JackSchmidt/Archives/2010/06

PSL(2,p), p = 5, 7, 11
Hi Jack,

Hope you’ve been well!

I recently learned something, which might be of interest if you’ve not heard of it or if it’s slipped your mind.

So do PSL(2,5), PSL(2,7), and PSL(2,11) mean anything to you?

Well, they’re the 3 (simple) PSL(2,p) groups that act on p points (non-trivially), which has been known for as long as we’ve known of these groups (i.e., since Galois), and now Wikipedia knows it too.

Some of these can be realized in nice geometric ways (via biplane geometries), but more subtly, these groups are another example of a McKay correspondence (see, the action on p points corresponds to a decomposition as sets $$A_4 \times Z_5,$$ $$S_4 \times Z_7,$$ and $$A_5 \times Z_{11},$$ which you’ll recognize as the Platonic solid groups), and thus correspond to $$\tilde E_6, \tilde E_7, \tilde E_8,$$ and much else besides – the (triple-covered) Fischer group 3.Fi24', the (double-covered) baby monster 2.B and the monster M, (extended monstrous moonshine), the 27 lines on a cubic surface, etc.

There are a few references too – this doesn’t seem terribly well-understood, and rather exceptional and pretty, so I thought you’d like it.
 * —Nils von Barth (nbarth) (talk) 06:20, 5 June 2010 (UTC)