# 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:

 $\displaystyle \map \alpha {\gen {A, B} }$ $=$ $\displaystyle \set {h N: h \in \gen {A, B} }$ $\displaystyle$ $=$ $\displaystyle \set {h N \in \gen {A / N, B / N} }$ $\displaystyle$ $=$ $\displaystyle \gen {\map \alpha A, \map \alpha B}$

$\blacksquare$