On superstable groups with residual properties

Given a pseudovariety C, it is proved that a residually-C superstable group G has a finite series G(0) <= G(1) <= (...) <= G(n) = G such that G(0) is solvable and each factor G(i + 1)/G(i) is in C (0 <= i <= n - 1). In particular, a residually finite superstable group is solvable-by-finite, and if it is omega-stable, then it is nilpotent-by-finite. Given a finitely generated group G, we show that if G is w-stable and satisfies some residual properties (residual solvability, residual finiteness,...), then G is finite.