On a notion of breadth in the sense of Frobenius

Given a finite group $G$ and an integer $e \ge 1$ dividing the order of $G$, the size of the set $L_e(G)=\{x \in G \mid x^e=1\}$ was studied originally by Frobenius, in order to find restrictions on the structure of $G$. The aim of the present paper is to classify groups by $\mathbf{B}(G)=\max \{ \frac{|L_e(G)|}{e} \mid e \in \mathrm{Div}(\exp(G)) \}$, where $\mathrm{Div}(\exp(G))$ is the set of all divisors of the exponent $\exp(G)$ of $G$. We will show general statements regarding the center and the central quotient of $G$, by looking at $\mathbf{B}(G)$. This improves some recent contributions of W. Meng and J. Shi in [On an inverse problem of Frobenius' theorem, Arch. Math. (Basel) 96 (2011), 109-114].