Abstract: Let G be an algebraic group, X a generically free G-variety, and K=k(X)^G. A field extension L of K is called a splitting field of X if the image of the class of X under the natural map H^1(K, G) ---> H^1(L, G) is trivial. If L/K is a (finite) Galois extension then Gal(L/K) is called a splitting group of X. We prove a lower bound on the size of a splitting field of X in terms of fixed points of nontoral abelian subgroups of G. A similar result holds for splitting groups. We give a number of applications, including a new construction of noncrossed product division algebras.