Abstract: This is a note I wrote for the open problems chapter of the Proceedings of the Classical Invariant Theory Workshop held at Queens College in Kingston, Ontario in April, 2002. For an algebraic group G, defined over an algebraically closed field of characteristic zero, there is a natural partial order on the set of G-actions on algebraic varieties: X >= Y if there exists a dominant G-equivariant rational map (i.e., a compression) from X to Y. Alternatively, one can consider regular, rather than rational, compressions. In this note I propose to study this partial order in the case where G is a finite group. In particular, I am interested in describing the minimal elements (I call them incompressible G-varieties).