Nilpotent residual of fixed points

Abstract

Let q be a prime and A a finite q-group of exponent q acting by automorphisms on a finite $$q'$$ -group G. Assume that A has order at least $$q^3$$ . We show that if $$\gamma _{\infty } (C_{G}(a))$$ has order at most m for any $$a \in A^{\#}$$ , then the order of $$\gamma _{\infty } (G)$$ is bounded solely in terms of m and q. If $$\gamma _{\infty } (C_{G}(a))$$ has rank at most r for any $$a \in A^{\#}$$ , then the rank of $$\gamma _{\infty } (G)$$ is bounded solely in terms of r and q.