Fri Nov 3

Transitive and simply transitive G-sets. G-equivariant maps and isomorphisms. G acting on a disjoint union of G-sets.
Theorem. Every G-set is G-equivariantly isomorphic to a disjoint union of transitive G-sets.
Free actions.
Theorem. If G acts freely on X finite, then |X| = |G| |X/G|.

Scary example: X = circle, G acts by antipode, X/G a circle again. Then the bijection used in the above theorem is not continuous. (Of course, continuity isn't very relevant when X is finite.)

Next time: we'll classify transitive G-sets and describes all morphisms between them.