This is an old exam question I'm trying to solve:
Having a group $G$ and $H$ a normal subgroup of $G$ with order $n$ and taking $g$ in $G$ to be such that $gH$ has order $m$ in $G/H$, I wish to prove that $\langle g \rangle H$ is a subgroup of $G$ of order $mn$.
My attempt:
If it's a subgroup, I can totally prove it has order $mn$. But I can't make it to prove that it is a subgroup for some reason. I guess I'll need to use the fact that h is closed under conjugacy but even with that I haven't made it... Can you help me? Any hint would be awazing! Thank you!