Uncountable saturated structures have the small index property

We prove the following theorem. Let m be an uncountable saturated structure of cardinality λ = λ^<λand assume that G is a subgroup of Aut (m) whose index is less than or equal to λ. Then there exists a subset A of cardinality strictly less than λ such that every automorphism of m leaving A pointwise fixed is in G.