MATH.8:4.4 - Explain duplicates and the limit of one orbit

The transformations fixing x form its stabilizer H_x={h in H | h*x=x}. Two transformations give the same answer precisely when the composition of one with the other’s inverse fixes x:

h1*x=h2*x exactly when (h2^-1*h1)*x=x.

Choose one transformation h0 in H. All transformations returning the answer h0*x have the form h0*k with k in H_x: composing with k leaves x unchanged, and any h giving that answer satisfies h0^-1*h in H_x. These sets of transformations are called the left cosets of the stabilizer. They partition H, and each contains as many transformations as H_x, since multiplication by h0 is reversible. Therefore, for finite H, the number of distinct answers is |H|/|H_x|. Use this count when it helps construct or check the requested family; explicit comparison can be simpler for a small example.

One orbit contains exactly the answers reachable from its starting member under H. To claim all solutions, establish that every solution belongs to a represented orbit. This can use a complete finite classification, a mathematical reduction, or an argument that H acts transitively on the solution set, meaning that every solution is reachable from the starting solution. Finding no new member in the current orbit proves its closure, not that another orbit is absent.

When another solution lies outside the family, use it as the starting member of another orbit. A property unchanged by every transformation can show that two candidates cannot belong to the same orbit. Such a property can also suggest what a wider transformation group would need to change.