The structure of a finite group and the maximum $\pi$-index of its elements

TL;DR

This paper investigates how the maximum $ ext{pi}$-index of elements in a finite group constrains the structure of its factor group over the center, linking element indices to group-theoretic properties like Frattini length and $ ext{pi}$-length.

Contribution

It establishes bounds on the Frattini length and $ ext{pi}$-length of the quotient group $G/Z(G)$ based on the maximum $ ext{pi}$-index of elements in $G$, revealing structural restrictions.

Findings

Proves $ ext{phi}_p(G/Z(G)) \,\leq\, \epsilon_p(G)$ for finite groups.

Shows $l_\pi(G/Z(G)) \leq \epsilon_\pi(G)$ for $\pi$-separable groups.

Links element index properties to the structural parameters of the quotient group.

Abstract

Given a set of primes $\pi$, the $\pi$-index of an element $x$ of a finite group $G$ is the $\pi$-part of the index of the centralizer of $x$ in $G$. If $\pi=\{p\}$ is a singleton, we just say the $p$-index. If the $\pi$-index of $x$ is equal to $p_1^{k_1}\ldots p^{k_s}$, where $p_1,\ldots,p_s$ are distinct primes, then we set $\exp_\pi(x)=k_1+\ldots+k_s$. In this short note, we study how the number $\epsilon_\pi(G)=\max\{\epsilon_\pi(x):x\in G\}$ restricts the structure of the factor group $G/Z(G)$ of $G$ by its center. First, for a finite group $G$, we prove that $\phi_p(G/Z(G))\leq\epsilon_p(G)$, where $\phi_p(G/Z(G))$ is the Frattini length of a Sylow $p$-subgroup of $G/Z(G)$. Second, for a $\pi$-separable finite group $G$, we prove that $l_\pi(G/Z(G))\leq\epsilon_\pi(G)$, where $l_{\pi}(G/Z(G))$ is the $\pi$-length of $G/Z(G)$.