Monday Nov 6, Wednesday Nov 8, Monday Nov 13

k-transitivity and the Mathieu groups (for culture only).
Every transitive G-set is isomorphic to a coset space G/H.
Thm. Let G act on X finite. Then |X| = sum |G/Stab_i|, the stabilizers appearing.
Example: Let S_n act on the power set. Then 2ⁿ = sum (n choose k).
Cor. |G| = |X| |Stab|. Ex. |Rot(cube)| = #faces * (rotations of a face) = 6*4.

When is there a map from G/H -> G/K?
Classifying such. If H=K is finite, then they're automatically invertible.
Ex. Let Rot(cube) act on unordered pairs of vertices.
Then the orbits correspond to distance 1, distance 2, distance 3.
There is a Rot(cube)-equivariant map from {distance 1} -> {distance 3}, but none between any other pair of distinct orbits.

No class Friday (Veteran's Day, observed).

Let |G|=pⁿ act on X. Then |X| = |X^G| mod p.
Ex. G = Z_p acting on Fun(G,{1,...,n}) gives n^p = n mod p.
Center of a group.
Conjugacy classes in S_n.
In A_n they're more complicated: A₅ example.