Finite groups admitting a coprime automorphism satisfying an additional polynomial identity

It is known that a finite group with an automorphism phi of coprime order has a soluble radical of (vertical bar phi vertical bar, vertical bar C-G(phi)vertical bar)-bounded Fitting height and index. We extend this classical result as follows. Let f (x) = a(0) + a(1) center dot x + center dot center dot center dot + a(d) center dot x(d) is an element of Z[x] be a primitive polynomial, and let G be a finite group with an automorphism phi of coprime order satisfying g(a0) center dot phi(g)(a1) center dot center dot center dot phi(d) (g)(ad) = 1 for all g is an element of G. Then the soluble radical of G has (d, vertical bar C-G(phi)vertical bar)-bounded Fitting height and index. The bounds are made explicit and are particularly good for small values of the degree d.