A semi-automorphism of a group G is a 1-1 mapping, X, of G onto itself such that 0(aba) =4(a)o(b)4(a) for all a, bEG. The nature of such mappings, in the special cases when G is the symmetric or alternating group (finite or infinite) and in a few other examples, was determined by Dinkines [I], who showed they must be automorphisms or anti-automorphisms. Her proof was rather computational in character. In her paper she conjectured that a semi-automorphism of a simple group is either an automorphism or an anti-automorphism. In this paper we prove this result for a wide class of simple groups, finite or infinite. In the process of doing so we are led to a simplified, and somewhat more conceptual, proof of Dinkines's results. In the body of this paper q will denote a semi-automorphism of the group G. We begin with