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:

$\displaystyle \bigcap_{a \mathop \in G} H^a = \set e$

where $H^a$ denotes the conjugate of $H$ by $a$.


Proof


Sources