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.

Comment: 7 pages

• 20D20 - Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
• 20D60 - Arithmetic and combinatorial problems involving abstract finite groups
• 20E45 - Conjugacy classes for groups

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