We prove for many groups considered in classical mathematics (Chevalley
groups over infinite fields, connected perfect linear algebraic groups,
infinite permutation and infinite dimensional general linear groups), a model
theoretical phenomenon called absolutely connectedness. Namely, G is absolutely
connected if for an arbitrary first order structure on G, working in a
saturated extension, G does not have any proper definable, type definable or
invariant under structure automorphisms subgroups of bounded index i.e.
G=G^0=G^00=G^{\infty}.