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 $\set {A, B}$.

From the proof of the Correspondence Theorem:

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

Then:

$\ds \map \alpha {\gen {A, B} }$

$=$

$\ds \set {h N: h \in \gen {A, B} }$

$\ds $

$=$

$\ds \set {h N \in \gen {A / N, B / N} }$

$\ds $

$=$

$\ds \gen {\map \alpha A, \map \alpha B}$

$\blacksquare$

1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Proposition $7.16 \ \text{(ii)}$