The notion of a strongly determined type over A extending p is introduced,
where p is an element of S(A). A strongly determined extension of p over A
assigns, for any model M superset of or equal to A, a type q is an element
of S(M) extending p such that, if (c) over bar realises q, then any element
ary partial map M --> M which fixes acl(eq)(A) pointwise is elementary over
(c) over bar. This gives a crude notion of independence (over A) which ari
ses very frequently. Examples are provided of many different kinds of theor
ies with strongly determined types, and some without. We investigate a noti
on of multiplicity for strongly determined types with applications to 'invo
lved' finite simple groups, and an analogue of the Finite Equivalence Relat
ion Theorem. Lifting of strongly determined types to covers of a structure
(and to symmetric extensions) is discussed, and an application to finite co
vers is given. (C) 1999 Elsevier Science B.V. All rights reserved.