Finite groups admitting some coprime operator groups

Abstract

Let $G$ be a finite group, with a finite operator group $A$,satisfying the following conditions:(1)~$(\vert G \vert, \vert A \vert)=1$;(2)~there exists a natural number $m$ such that for any $ \alpha,\beta ın A^{\sharp}$ we have:$[\,C_G(\alpha),\underbrace{C_G(\beta),\dots,C_G(\beta)}_{m}\,]=\{1\}$;(3)~$A$ is not cyclic. We prove the following:(1)~If the exponent $n$of $A$ is square-free, then $G$ is nilpotent and its class isbounded by a function depending only on $m$ and $\lambda(n)$ ($=n$).(2)~If $Z(A)=\{1\}$ and$A$ has exponent $n$, then $G$ is nilpotent and its class isbounded by a function depending only on $m$ and $\lambda(n)$.