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.