MATH.8:4 - Solution

Specify the actions → preserve the solution relation → transform a known solution → construct its orbit → establish the family’s reach → return the needed answer.

MATH.8:4.1 - Specify the problem and the transformations

Write the solution condition as S(d,x): candidate x solves the problem with data d. State the data that are held fixed, the domain of candidates and the equality used to distinguish answers. An equation, a feasibility condition or a minimum under a stated criterion can provide S.

Give the transformations and their actions on d and x. For a group G, the action satisfies e*x=x and (g*h)*x=g*(h*x); the same rules apply to the data action. Here e is the identity transformation, and multiplication in G means composition, with h applied first. Every g has an inverse. A supplied family of permutations can make these rules immediate.

If the group is given by generating transformations, include their inverses and retain the relations they must satisfy. MATH.5 can construct an action from generator images that respect those relations. Listing generators without their action leaves the proposed symmetry undecided.

For a fixed datum d, retain only transformations with g*d=d. These form its stabilizer: the subgroup of transformations that leave that datum unchanged. Include compositions when finding this subgroup: the two-mark case in :5.2 is fixed by a half-turn although a single turn changes the marks. A larger group can relate different data instances, which is a different useful calculation.

MATH.8:4.2 - Establish the solution-preserving relation

Show that S(d,x) implies S(g*d,g*x) for every transformation and candidate in the claimed range. Applying the same implication to the inverse gives the converse. The transformation therefore pairs the two solution sets.

For an equation, substitute the transformed variables and data. For several constraints, preserve each one used by the solution. An optimization result needs both transformed feasibility and the corresponding criterion: if feasible candidates are paired bijectively and J_(g*d)(g*x)=J_d(x), a better transformed candidate would return a better original candidate. Thus a minimizer transfers.

Preservation for generating transformations and their inverses extends to every finite composition by applying the implications successively. Use MATH.4’s finite-construction argument when that extension needs explanation. A check on a few candidate values establishes only those cases unless a general argument is also available.

If a transformation fails, retain the first changed condition and decide whether the changed problem is useful. MATH.6 can exhibit the failure; FPF C.29 relates the mathematical correspondence to another subject when one is involved.

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.

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.

MATH.8:4.5 - Use an orbit representative without losing the requested result

A quantity constant on each orbit can be calculated from any representative. To define additional operations on orbit classes, use MATH.2’s representative-independence condition; an orbit partition by itself only supplies classes.

When a problem is solved using representative data d0 and solution x0, retain a transformation g with g*d0=d for the requested data d. Return g*x0. If two transformations satisfy g1*d0=g2*d0=d, an answer independent of that choice requires g1*x0=g2*x0. A disagreement identifies information that the representative alone does not supply.

MATH.13 supplies the fixed-point restriction on a unique solution and the test for an impossible equivariant choice. Those questions require the output action and, for the unique-solution deduction, a justified uniqueness premise. Generating a solution set here does not select one member from it.

Stop with the requested related solution, complete orbit, orbit classification or identified failure of preservation. When data, constraints or the output change, reopen the affected action and preservation argument before reusing the family. A more costly enumeration or symmetry computation is useful only if it can improve the receiving result.