ON AN INVERSE PROBLEM TO FROBENIUS' THEOREM II

Let G be a finite group and e a positive integer dividing vertical bar G vertical bar, the order of G. Denoting Le(G) = {x is an element of G vertical bar x(e) = 1}. Frobenius proved that vertical bar L-e(G)vertical bar = ke for some positive integer k >= 1. Let k(G) be the upper bound of the set {k vertical bar vertical bar L-e(G)vertical bar = ke, for all(e) vertical bar vertical bar G vertical bar}. In this paper, a complete classification of the finite group G with k(G) = 3 is obtained.