借力“油菜花节” 汉中签订23个重点投资项目

Question

Let $G$ be a group and $H < G$ is a subgroup. If $\Gamma = \{gH \ | \ g \in G\}$ then $G$ act on $\Gamma$ by "left inverse" multiplication like $(gH) \circ t = t^{-1}gH$. But what happens when we multiply on right, i. e. $gH \circ t = gHt$? When does such rule determine group action on $\Gamma$?

If $H$ is normal in $G$ we are clearly got an action $G$ on $\Gamma$. Also if $H$ isn't normal in $G$ then $N_{G}(H)$ acts on $\Gamma$ by right multiplication. Here I stacked... when does right multiplication of left coset is left coset?

Answer 1

Your question is: for which $t\in G$ do we have$$\forall g\in G\ \exists g'\in G\quad gHt=g'H.$$ This is equivalent to:$$\exists g'\in G\quad Ht=g'H.$$ It requires $t\in g'H$, so $tH=g'H$ and your condition simplifies to $$Ht=tH,$$i.e. $t\in N_G(H)$.