If TV is a group and £ is a group of operators on N then write dE(N) for the minimum number of elements needed to generate A' as an ¿"-group. It is shown that if N is a normal subgroup of E and E acts on N by conjugation, then <¡eÍn) = dE(N/N') if dE(N) is finite and there does not exist an infinite descending series of ¿"-normal subgroups N' = C0 > C, > ■ ■ -with each C¡/C¡+¡ perfect. Both these conditions are, in general, necessary.