On a problem of Frobenius

Let G be a finite group and let n be a natural integer. We define n_G = (n, |G|) and L_n(G) = {g ε{lunate} G| gⁿ = 1}. We shall write G ε{lunate} F_n if |L_n(G)| = n_G. Frobenius conjectured that if G ε{lunate} F_n, then L_n(G) is a normal subgroup of G (or, in short, G is n-closed). A weaker conjecture of Frobenius states that if H ε{lunate} F_n for every subgroup H of a finite group G (including G itself), then G is n-closed. This weaker conjecture is proved in this article for natural numbers n not divisible by 4. In fact, our result is more general than the weaker Frobenius conjecture.