On an inverse problem to Frobenius' theorem

Let G be a finite group and e a positive integer dividing vertical bar G vertical bar, the order of G. Denoting L(c)(G) = {x is an element of G vertical bar x(c) = 1}, Frobenius proved that vertical bar L(e)(G)vertical bar = ke for a positive integer k >= 1. In this paper, we give a complete classification of finite groups G with vertical bar L(e)(G)vertical bar <= 2e for every e dividing vertical bar G vertical bar.