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:
\(\ds \map \alpha {\gen {A, B} }\) | \(=\) | \(\ds \set {h N: h \in \gen {A, B} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \set {h N \in \gen {A / N, B / N} }\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \gen {\map \alpha A, \map \alpha B}\) |
$\blacksquare$
Sources
- 1996: John F. Humphreys: A Course in Group Theory ... (previous) ... (next): Chapter $7$: Normal subgroups and quotient groups: Proposition $7.16 \ \text{(ii)}$