Condition for Group to Act Effectively on Left Coset Space
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Theorem
Let $G$ be a group whose identity is $e$.
Let $H$ be a subgroup of $G$.
Then $G$ acts effectively on the left coset space $G / H$ if and only if:
- $\ds \bigcap_{a \mathop \in G} H^a = \set e$
where $H^a$ denotes the conjugate of $H$ by $a$.
Proof
$G$ acts effectively on the left coset space $G / H$ if and only if $a H \mapsto g a H$ is faithful, if and only if:
| \(\ds \forall g \in G: \forall a H \in G / H: \, \) | \(\ds g a H = a H\) | \(\implies\) | \(\ds g = e\) | Definition of Faithful Group Action | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall g \in G: \forall a \in G: \, \) | \(\ds a^{-1} g a \in H\) | \(\implies\) | \(\ds g = e\) | Left Cosets are Equal iff Product with Inverse in Subgroup | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall g \in G: \forall a \in G: \, \) | \(\ds g \in a H a^{-1}\) | \(\implies\) | \(\ds g = e\) | ||||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \forall g \in G: \forall a \in G: \, \) | \(\ds g \in H^a\) | \(\implies\) | \(\ds g = e\) | Definition of Conjugate of Group Subset | |||||||||
| \(\ds \leadstoandfrom \ \ \) | \(\ds \bigcap_{a \mathop \in G} H^a\) | \(=\) | \(\ds \set e\) | Definition of Intersection of Family |
$\blacksquare$
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $2$: The Sylow Theorems: $\S 53 \delta$