On existence of normal p-complement of finite groups with restrictions on the conjugacy class sizes

The greatest power of a prime $p$ dividing the natural number $n$ will bedenoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite groupand $p$ is a prime. We prove that if there exists an integer $\alpha>0$ suchthat $Ind_G(a)_p\in \{1,p^{\alpha}\}$ for every $a$ of $G$ and a $p$-element$x\in G$ such that $Ind_G(x)_p>1$, then $G$ includes a normal $p$-complement.Comment: 7 page