Generator of Quotient Groups

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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:

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

$\blacksquare$


Sources