A partial analogue of Borel's Fixed Point Theorem for finite p-groups

Borel's Fixed Point Theorem states that a solvable connected algebraic group G 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 in Borel's Theorem, the method of proof yields applications to finite p-groups. They show that in many cases, a family of subgroups of a finite p-group must contain at least one normal subgroup. Here, we obtain some applications of this type.