Are $g_1$ and $g_2$ elements of $\langle g_1g_2\rangle$ when $g_1$ and $g_2$ commute and have co-prime orders?

Question

Consider a finite group $G$. Let $g_1$ and $g_2$ two elements of $G$ that commute and have co-prime orders. Are $g_1$ and $g_2$ elements of $\langle g_1g_2\rangle$ ?

Answer 1

Let $o(g_1)=m$ and $o(g_2)=n$, so gcd$(m,n)=1$. By Bézout's Theorem we can find $k,l \in \mathbb{Z}$ with $1=km+ln$. It follows that $g_1=g_1^{ln}$ and $g_2=g_2^{km}$. Hence $(g_1g_2)^{ln}=g_1$ and $(g_1g_2)^{km}=g_2$.