An analogue of Borel’s Fixed Point Theorem for finite p-groups

Borel’s Fixed Point Theorem states that a solvable connected algebraic group G acting on a non-empty complete variety V must have a fixed point. Thus, if V consists of subgroups of G, and G acts on V by conjugation, then some subgroup in V is normal in G.

Although G is infinite or trivial here, we can use the method of proof to obtain applications to finite p-groups. We plan to discuss some applications and some open problems. No previous knowledge of algebraic groups is needed.