# Condition for Group to Act Effectively on Left Coset Space

## 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

$\blacksquare$