Orders of Cosets

Question

Let H be a normal subgroup of G and for any $g \in G$ that $|g|=n$ and $|G/H|=m$. Suppose the $gcd(n,m)=p$ where $p$ is prime. Show that for any $ a\in gH$, then $a^p\in H$.

Could I get some help on this? I know that the order of $|gH|$ divides both $|G/H|$ and $|g|$.

Answer 1

Hint: Bezout's Lemma guarantees the existence of integers $x$ and $y$ with $xn+ym=p$. Hence $a^{p}=(a^y)^m$.