MATH.8:4.3 - Generate related solutions and remove repetitions

From a known solution x, calculate g*x and return it with the data g*d. To obtain solutions of the original fixed problem, use its data stabilizer from :4.1.

For a chosen group H acting on the same fixed problem, define the orbit:

H*x={h*x | h in H}.

All its members solve that problem by :4.2. Compare the resulting objects using the problem’s equality; distinct transformation expressions can give equal answers.

With a finite list of generating transformations, a finite orbit and decidable equality, generate the orbit by closure. Start with x. Apply each generating transformation and its inverse to every newly found member, adding only previously absent results. Once every stored member has been processed and no new one appears, the set is closed under the generators and inverses. Every finite word in them stays in that set, so it is the whole generated orbit.

This procedure terminates when the reached orbit is finite and the stated operations return. For an infinite orbit, a formula such as {h*x | h in H} with a usable parameterization can be the result. A stopped enumeration without closure supplies only the reached subset.