I am currently working through Lang's Algebra. I am rather confused by what seems to be a trivial point. In a lemma preceding the proof of the Sylow Theorem (which is essentially Cauchy's Theorem), lemma 6.1, he proves that if a finite abelian group has an exponent $n$ then its order divides some power of $n$. I am comfortable with this fact. However, he immediately used this to show that all such groups with order $np$, for $p$ prime, have an element of period $p$. This seems very much like a non-sequitur to me. What am I missing?
Let’s see. If there’s no element of order $p$, mustn’t the exponent be prime to $p$? Granting that, let the exponent be $m$, but then $np | m^r$, looks like a contradiction.