Separability properties came from residual properties which, according to Gruenberg [23], were first suggested by P. Hall. In the wider applications of these concepts D. Solitar preferred using the term separable. Therefore, we define S-separability of a group as follows:

(1) If S is a subset of a group G, then we say G is S-separable if for each g ∈ G\S and g = 1, there exists a normal subgroup Ng , depending on g, of finite index in G such that in G = G/Ng , g ∈ S. That is S is closed under the profinite topology of G.