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

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 $p_1^{k_1}\dots p^{k_s}$, where $p_1,\dots ,p_s$ are distinct primes, then we set ${\exp }_{\pi }(x)=k_1+\dots +k_s$. 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 ${\varphi }_p(G/Z(G))\le {\epsilon }_p(G)$, where ${\varphi }_p(G/Z(G))$ is the Frattini length of a Sylow $p$-subgroup of $G/Z(G)$. Second, for a $\pi$-separable finite group $G$, we prove that $l_{\pi }(G/Z(G))\le {\epsilon }_{\pi }(G)$, where $l_{\pi }(G/Z(G))$ is the $\pi$-length of $G/Z(G)$.