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 $q^{\prime }$-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 $q$.