the connected component of identity in G . Conversely, if G is a Lie group with an involutive automorphism E and H is a subgroup satisfying Go a H c GE, then H is a closed subgroup of G and the coset space G/H is an analytic manifold; furthermore, the canonical invariant affine connection of G induces an affine connection on M = G / H, which renders M an affine symmetric space with symmetries derived from E in an obvious manner. Such an affine symmetric space will be denoted by (G/H,X) or simply by G/H. The discussion given in the preceding paragraph shows that we may restrict our study of affine symmetric space to the case M = G / H, where G is a connected Lie group. We may, and we shall, further assume that GIH is effective in the sense that H contains no nontrivial normal subgroup of G; for if we take G =AO(M), then any normal subgroup H' c H induces identity transformation on M. In general, if G is a group of affine transformation of a connected, affinely