The diameter of a Lascar strong type

We prove that a type-definable Lascar strong type has finite diameter. We answer also some other questions from [1] on Lascar strong types. We give some applications on subgroups of type-definable 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 ∈ C let a0Θa1 iff 〈a0, a1〉 extends to an indiscernible sequence 〈an, n < ω〉. We define a distance function d on C. Namely, d(a, b) is the minimal natural number n such that for some a0 = a, a1, . . . , an−1, an = b we have a0Θa1Θ . . . an−1Θan. If no such n exists, we set d(a, b) =∞. The transitive closure Ls ≡ of Θ (denoted also by EL) is the finest bounded invariant equivalence relation on C, its classes are called Lascar strong types. So a Ls ≡ b ⇐⇒ d(a, b) < ∞. bd ≡ (denoted also by EKP ) is the finest bounded typedefinable equivalence relation on C. For details see e.g. [1]. So bd ≡ is coarser than Ls ≡ and each bd ≡-class is a union of some number of Lascar strong types.