Abstract: | Let A be an elementary abelian group of order at least p 3 acting on a finite p′-group G that is soluble with derived length d. Assume that γ c (C G (a)) has exponent dividing m for any a ∈ A #. It is proved that there exist {p, d, c, m}-bounded numbers c 1 and m 1 such that γ c 1 (G) has exponent dividing m 1. |