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Covering Conditions on Commutators and Bounded Derived Structure in Groups

A group satisfies the (k,n)-covering condition for commutators when its commutators can be covered by at most k translates of the set of elements whose conjugacy class has size at most n. Under this condition the derived subgroup contains a characteristic subgroup whose index and whose own derived subgroup both have order bounded in terms of k and n alone, so the group is close to being nilpotent of class two. The result unifies and extends bounded-conjugacy-class (BFC) theorems in group theory.