I'm trying to prove the correspondence theorem differently than my professor did. Is this valid?
Theorem: Let $G$ be a group and $N \le G$ be a normal subgroup. Then there is a $1-1$ correspondence between subgroups of $G/N$ and subgroups of $G$ containing $N$.
Proof: Let *Sub*$ (G/N)$ and *Sub*$(G)$ denote the set of subgroups of $G/N$ and subgroups of $G$ containing $N$ respectively. Define the map $f$: *Sub*$(G)$ $\to$ *Sub*$(G/N)$ by $f(H) = H/N$. This map is injective since if $f(H) = f(H')$, by the fact that the cosets partition a group,
$H = \bigcup_{a \in H} aN = \bigcup_{a \in H'} aN = H'$
To show that $f$ is surjetive, let $K \le G/N$ be a subgroup. We show that $V = \cup_{\alpha \in K} \alpha$ is a subgroup. Let $x,y \in V$. It follows that $x = an $ and and $y = bn'$ where $a,b \in G$ and $n, n' \in N$. So $xy = (an)(bn') = abn''n$ for some $n'' \in N$ , and hence $xy \in V$. Furthermore, for any $an \in V$ $(an)^{-1} =n^{-1}a^{-1} \in V$, so $V$ is a subgroup which clearly contains $N$, and $f(V) = K$.