Transcription of Math 412. The Orbit Stabilizer Theorem
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Math 412. The Orbit Stabilizer Theorem
(c)Karen E. Smith 2018 UM math Deptlicensed under a Creative CommonsBy-NC-SA International 412. The Orbit Stabilizer TheoremFix an action of a groupGon a setX. For each pointxofX, we have two important concepts:DEFINITION: Theorbitofx Xis the subset ofXO(x) :={g x|g G} : Thestabilizerofxis the subgroup ofGStab(x) ={g G|g x=x} :If a finite groupGacts on a setX, then for everyx X, we have|G|=|O(x)| |Stab(x)|.A. LetD4be the symmetry group of the square. Consider the natural action ofD4on the setR2byrotations and reflections of the whole space.(1) Complete the following chart which records, for different points of the setR2, the Orbit , Stabilizer ,and cardinalities of each. Use the notation{e, r, r2, r3, x, y, d, a}.(2) Now verify the Orbit Stabilizer Theorem for each of the five points in your chart. Easy: the numberof elements in the Orbit times the number of elements in the Stabilizer is the same, always 8, foreach THESTABILIZER OF EVERY POINT IS A a groupGacts on a setX.
The orbit of a triangle is the set of all congruent triangles. The stabilizer of an equilateral triangle consists of all the rigid motions which fix the triangle (as a set). This is D 3, basically by definition of D 3. The stabilizer of an isoceles triangle is much smaller: we only have reflections across one axis, so it is a 2-element
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