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

## TL;DR

This paper proves that finite groups with conjugacy class sizes restricted to certain p-power conditions necessarily have a normal p-complement, under specific divisibility assumptions.

## Contribution

It establishes a new criterion for the existence of a normal p-complement based on conjugacy class size restrictions in finite groups.

## Key findings

- If conjugacy class sizes' p-part is 1 or p^α for all elements
- Existence of a p-element with class size divisible by p
- Group has a normal p-complement under these conditions

## Abstract

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.