COMBINATORICS AND THE SYLOW THEOREMS
The Sylow theorems appear at the very start of group theory. If you look (and we will) you'll see that the proofs are 'just simple combinatorics'. This connection continues. In this talk I'll focus on the Sylow subgroups of the Symmetric group S_n. These have a simple description as 'chandelier graphs'. Their characters (and conjugacy classes have a simple description as '2-3 trees'. The double cosets don't (yet) have any kind of description but it you could find one, it would help prove things like the 'Alperin weight conjecture'. I'll try to explain all of this 'in English'. There is a lot of probability and combinatorics along the way.