Suppose H & K are 2 subgroups of a group G. May I get help on the 3 related problems:
1.Show that the set of maps x->hxk h in H, k in K is a group of transformations of the set G.
2. Show that the orbit of x relative to this group is the set HxK={hxk|h in H, k in K}
3. Show that if G is finite then |HxK|=|H|[K:x-1Hx intersection K]=
|K|[H:xKx-1 intersection H]
I'll appreciate you help.