symmetrisets

Symmetrisets are a mathematical concept used to describe subsets that remain unchanged under a prescribed set of symmetries. In formal terms, let X be a set and G a group acting on X by a collection of transformations (a group action). A subset S of X is called a symmetriset with respect to this action if g(S) = S for every element g in G. Equivalently, S is a union of G-orbits and is invariant under the entire symmetry group.

Examples illustrate the idea. In the Euclidean plane, with the dihedral group D4 acting on the plane, the unit circle is a symmetriset because it is preserved under all rotations and reflections in D4. The unit square is another symmetriset for the same group. Trivial examples include the empty set and the whole space X, which are invariant under any group action.

Core properties follow from the definition. The collection of symmetrisets with respect to a fixed G forms a lattice under inclusion, since intersections and unions of invariant sets are themselves invariant when the same group action is considered. If S1 and S2 are symmetrisets for G, then their intersection is a symmetriset, and their union is a symmetriset as well, provided the action of G is consistent across the union.

Relation to other concepts is direct: symmetrisets are a form of G-invariant sets in invariant theory, closely linked to orbits, stabilizers, and fixed-point phenomena. Applications appear in crystallography, pattern recognition, computer graphics, and the study of symmetric patterns.

History and terminology notes: the term “symmetriset” appears in some texts as a descriptive name for G-invariant subsets; some authors prefer “G-invariant set” or “invariant subset.” See also symmetry group, group action, invariant set, orbit, fixed point.