Influence of Conjugacy Class Sizes of Some Elements on the Structure of a Finite Group

Abstract

Let $G$ be a finite group. For every element $x\in G$, the set $\{x^g=g^{-1}xg: g\in G\}$ is called the conjugacy class of $x$ in $G$ and is denoted by $x^G$. The conjugacy class size of $x$ in $G$ is denoted by $|x^G|$ and is equal to $|G:C_G(x)|$. An element $y$ of $G$ is said to be primary or biprimary if the order of $y$ is divisible by exactly one or two distinct primes. For a positive integer $n$ and a prime $p$, if $e>0$ is an integer such that $p^e$ divides $n$ and $p^{e+1}$ does not divide $n$, then $p^e$ is called the $p$-part of $n$. Let $p$ be a prime divisor of $|G|$ such that $(p-1,|G|)=1$. We prove that $G$ is solvable and $p$-nilpotent if the conjugacy sizes of all noncentral primary and biprimary elements in $G$ have the same $p$-part. On the other hand, suppose that $N$ is a normal subgroup of $G$. We write $\operatorname{cs}_G(N)=\{|x^G|:x\in N\}$. Suppose that $\operatorname{cs}_G(N)=\{1,n_1,n_2,\dots,n_t\}$, where $1<n_1<n_2<\cdots<n_t$. Denote by

$$M_N(G)=\langle x:x\in N,\,x^G=1\text{ or }n_1\rangle$$

a subgroup of $G$. We prove that if $C_G(F(G))\le F(G)$ and $[x,F(G)]$ is a normal subset of $F(G)$ for every $x\in N$ with $|x^G|=n_1$, then $M_N(G)$ is a nilpotent group with nilpotency class at most 2.