Generator of Quotient Groups

Theorem
Let $N \lhd G$ be a normal subgroup of $G$.

Let:
 * $N \le A \le G$
 * $N \le B \le G$

For a subgroup $H$ of $G$, let $\alpha$ be the bijection defined as:
 * $\map \alpha H = \set {h N: h \in H}$

Then:
 * $\map \alpha {\gen {A, B} } = \gen {\map \alpha A, \map \alpha B}$

where $\gen {A, B}$ denotes the subgroup of $G$ generated by $\set {A, B}$.

Proof
From the proof of the Correspondence Theorem:


 * $\map \alpha H \subseteq G / N$

Then: