A stronger form of Neumann's BFC-theorem

Given a group G, we write x^G for the conjugacy class of G containing the element x. A famous theorem of B. H. Neumann states that if G is a group in which all conjugacy classes are finite with bounded size, then the derived group G′ is finite. We establish the following result. Let n be a positive integer and K a subgroup of a group G such that |x^G| ≤ n for each x ∈ K. Let H=⟨K^G⟩ be the normal closure of K. Then the order of the derived group H′ is finite and n-bounded. Some corollaries of this result are also discussed.