Assume that G is a solvable group whose elementary abelian sections are all finite. Suppose, further, that p is a prime such that G fails to contain any subgroups isomorphic to Cp∞. We show that if G is nilpotent, then the pro-p completion map induces an isomorphism for any discrete -module M of finite p-power order. For the general case, we prove that G contains a normal subgroup N of finite index such that the map is an isomorphism for any discrete -module M of finite p-power order. Moreover, if G lacks any Cp∞-sections, the subgroup N enjoys some additional special properties with respect to its pro-p topology.

We investigate extensions of S. Solecki's theorem on closing off finite partial isometries of metric spaces [11] and obtain the following exact equivalence: any action of a discrete group Γ by isometries of a metric space is finitely approximable if and only if any product of finitely generated subgroups of Γ is closed in the profinite topology on Γ.

Natural algorithms to compute rational expressions for recognizablelanguages, even those which work well in practice, may produce very longexpressions. So, aiming towards the computation of the commutative image of arecognizable language, one should avoid passing through an expressionproduced this way.We modify here one of those algorithms in order to compute directly a semilinear expression for the commutative imageof a recognizable language. We also give a secondmodification of the algorithm which allows the direct computation of theclosure in the profinite topology of the commutative image. As anapplication, we give a modification of an algorithm for computing the Abelian kernel of a finite monoid obtainedby the author in 1998 which is much more efficient in practice.