I'm just confused about a somewhat simple fact about quotient groups. If we have:
$$H<G/N$$ is a subgroup of the quotient of a finite group $G$ by $N\trianglelefteq G$, and $|H|=n$. Can we necessarily conclude that $G$ has a subgroup of order $n$? I'm pretty sure this is false, in general.
Do we have that $$H\cong K/N$$ for some subgroup $K\leq G$, I think this is what my professor was saying. In that case, would the order of the cooresponding subgroup $K$ have $|K|=n\cdot |N|$? So that the existence of a subgroup $H$, like this necessarily implies there is a subgroup $H\leq G$ s.t. $|K|=n\cdot |N|$? Does anything change if $N$ is not normal in $G$?