Abstract
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 G| g n = 1}. We shall write G ϵ F n if | L n ( G)| = n G . Frobenius conjectured that if G ϵ 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 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.
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