We prove two results. (1) There is an absolute constant D such that for any finite quasisimple group S, given 2D arbitrary automorphisms of S, every element of S is equal to a product of D ‘twisted commutators’ defined by the given automorphisms. (2) Given a natural number q, there exist C = C(q) and M = M (q) such that: if S is a finite quasisimple group with |S/Z(S)| C, βj (j = 1, . . . , M ) are any automorphisms of S, and qj (j = 1, . . . , M ) are...