Nilpotent residual and Fitting subgroup of fixed points in finite groups

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 $m$. If the Fitting subgroup of $C_G(a)$ has index at most $m$ for any $a\in A^{\#}$, then the second Fitting subgroup of $G$ has index bounded solely in terms of $m$.

arXiv admin note: substantial text overlap with arXiv:1810.05663