Left actions of quotient maps

Question

I want to show that $G \times G/H \to G/H$, $(g,xH) \to gxH$ is a left action for $G/H=\{gH: g \in H\}$ where $G$ is a Lie group.

If I write, $e.gxh=gxh$ and $g(xh_1)(xh_2) \to g(xh_1xh_2)=(gxh_1)xh_2$, is it correct?

Answer 1

The Lie group structute is not needed here. Let $$\sigma: G \times G/H \longrightarrow G/H$$ $$(g_0,gH) \longmapsto (g_0g)H.$$ To show it is a (left) group action, you must verify:

1. $(\forall g \in G): \ \sigma(e,gH) = gH$;
2. $(\forall g_1, g_2, g \in G): \ \sigma(g_1,\sigma(g_2,gH)) = \sigma(g_1g_2, gH)$.

And in order to show this action is well-defined, you must lastly show that if $g'H= g''H$, then $$(\forall g \in G): \ \sigma(g,g'H) = \sigma(g,g''H).$$