Let G be af inite group. Then L s (G), the linear complexity of a suitable Wedderburn transform corresponding to G, is smaller than 2·| G| 2. We improve this trivial upper bound by showing that L s (G) is smaller than min {(s (T)− l (T))·| G|+ 7 q (T)·| G| 3 2, where the minimum is taken over all towers T=(G n> G n− 1>⋯> G 0={1}) of subgroups of G= G n, and where q (T)(resp. s (T)) is the maximum (resp. sum) of all indices [G i+ 1: G i] corresponding to T and l (T)= n is the length of the tower. A similar upper bound is valid for the linear complexity of the inverse of such a Wedderburn transform. For symmetric groups our technique yields the stronger estimate L s (S n)= 0 (| S n|· log 3| S n|).