On existence of normal p-complement of finite groups with restrictions
on the conjugacy class sizes
Authors: Ilya Gorshkov
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Ilya Gorshkov
The greatest power of a prime $p$ dividing the natural number $n$ will be
denoted by $n_p$. Let $Ind_G(g)=|G:C_G(g)|$. Suppose that $G$ is a finite group
and $p$ is a prime. We prove that if there exists an integer $\alpha>0$ such
that $Ind_G(a)_p\in \{1,p^{\alpha}\}$ 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.