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

cm:9294 - Communications in Mathematics, May 12, 2022, Volume 30 (2022), Issue 1 - https://doi.org/10.46298/cm.9294
On existence of normal p-complement of finite groups with restrictions on the conjugacy class sizes

Authors: 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.

Volume: Volume 30 (2022), Issue 1
Published on: May 12, 2022
Accepted on: April 5, 2022
Submitted on: April 3, 2022
Keywords: Mathematics - Group Theory