On Groups of Odd Order Admitting an Elementary 2-Group of Automorphisms

Abstract

Let $G$ be a finite group of odd order with derived length $k$. We show that if $G$ is acted on by an elementary abelian group $A$ of order $2^n$ and $C_G(A)$ has exponent $e$, then $G$ has a normal series $G=G_0\ge T_0\ge G_1\ge T_1\ge \cdots \ge G_n\ge T_n=1$ such that the quotients $G_i/T_i$ have $\{k,e,n\}$-bounded exponent and the quotients $T_i/G_{i+1}$ are nilpotent of $\{k,e,n\}$-bounded class.