On a notion of breadth in the sense of Frobenius

Given a finite group G and an integer e≥1 dividing the order of G, the size of the set Le(G)={x∈G|xe=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 B(G)=max⁡{|Le(G)|e|e∈Div(exp⁡(G))}, where 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 B(G). This improves some recent contributions of W. Meng and J. Shi in [7].