The diameter of a Lascar strong type

We prove that a type-denable Lascar strong type has nite diameter. We also answer some other questions from (1) on Lascar strong types. We give some applications on subgroups of type-denable groups. In this paper T is a complete theory in language L and we work within a monster model C of T. For a0;a1 2 C let a0 a1 i ha0;a1i extends to an indiscernible sequencehan;n < !i. We dene a distance function d on C by letting d(a;b) be the minimal natural number n such that for some a0 = a;a1;:::;an 1;an = b we have a0 a1 :::an 1 an. If no such n exists, we set d(a;b) =1. The transitive closure Ls of (denoted also by EL) is the nest bounded invariant equivalence relation on C; its classes are called Lascar strong types. So a Ls b, d(a;b) <1. Moreover, bd (denoted also by EKP) is the nest bounded type-denable equivalence relation on C. For details see e.g. (1).