# Group Actions and Normal Subgroups

In summary, the formula g(xH) = gxH defines an action of G on X. For H to be a normal subgroup of G, every orbit of the induced action of H on X must contain just one point. This means that for all h in H, the formula h(xH) = Hx holds true, which implies that xihxH belongs to H for all x in G and h in H.

## Homework Statement

Let H be a subgroup of G and let X be the set of left cosets of H in G.
Show that the formula g(xH) = gxH is an action of G on X.
Prove that H is a normal subgroup of G if and only if every orbit of the induced action of H on X contains just one point.

## The Attempt at a Solution

I've shown that the formula defines an action.
For the second part, since H is normal its left and right cosets are equivalent. Thus we consider h(xH) for h in H. So h(xH) = hxH = Hhx. But since h is in H this is the same as Hx. So for all h, the orbit contains only the point Hx.
However, the converse of the second part is what is giving me trouble. We know that each orbit contains only one element, but I'm not sure what else we can can from that.

H is a subgroup, so it includes e. For all h, h(xH)=e(xH)=xH. Left-multiplying by xi (x inverse) on both sides gives xihxH=H, so xihx belongs to H (for all x in G and h in H).