We interpret the basic notions of topological dynamics in the model-theoretic setting, relating them to generic types of definable group actions and their generalizations.

Abstract. We prove that a type-definable Lascar strong type has finite diameter. We also answer 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 a 0 , a 1 ∈ C let a 0 Θa 1 iff a 0 , a 1 extends to an indiscernible sequence a n , n < ω . We define a distance function d on C by letting d(a, b) be the minimal natural number n such that for some a 0 = a, a 1 , . . . , a n−1 , a n = b we have a 0 Θa 1 Θ . . . a n−1 Θa n . If no such n exists, we set d(a, b) = ∞. The transitive closure

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Let G be a group definable in a theory T . For every model M of T , the space S G (M ) of the complete G-types over M is a G M -flow. We compare the Ellis semigroups related to the flows S G (M ) and S G (N ) when M ≺ * N , focusing particularly on the groups into which the minimal left ideals in these semigroups split. In the case where T is an o-minimal expansion of the theory of reals and G is definably compact we show that these groups are isomorphic to the quotient group G/G 00 .

We propose a model-theoretic framework for investigating profinite groups. Within this framework we define and investigate small profinite groups. We consider the question if any small profinite group has an open abelian subgroup.