The structure of a finite group and the maximum $π$-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 $p_1^{k_1}\ldots p^{k_s}$, 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 $G/Z(G)$.