Remark (Recall). An action of on is a map
For , the orbit of is , and the stabilizer is . Then
Finite subgroups of = {rotations of }
Given , i.e., a real vector space with a dot product, we get the orthogonal group
Here : preserves lengths and angles. Inside, = orientation-preserving isometries ().
In dimension 2: , and
Given , there is a decomposition with , or , and (each subspace is invariant). So there is an orthonormal basis in which is block diagonal with and blocks.
the matrix of is
If : = linear operators which, in some orthonormal basis, have matrix with (e.g., ). All are rotations by some angle about an axis.
Given a subset , its symmetry group is
It could be infinite: if is a circle in the -plane, its symmetries are all rotations about the -axis, and the rotations by about lines in the -plane ().
But finite is more interesting.
Example N32.1. = a regular -gon in the -plane the symmetry group is (the dihedral group): rotations about the -axis and flips ( rotations about axes in the -plane). (The degenerate cases , a two-sided “lens”, and are drawn too.)
Example N32.2.: the cyclic group generated by the rotation by (e.g., the symmetries of a pyramid over a regular -gon).
Symmetries of regular polygons; regular polyhedra
Duals: for ,
Continuing: the dodecahedron (12 faces; drawn as a ball, but it’s not a circle!) and the icosahedron (20 faces), its dual.
Symmetries of the tetrahedron are given by ; cube: ; octahedron: also ; dodecahedron: ; icosahedron: .
Theorem N32.3. This is the complete list of finite subgroups of :
The poles of : the two points where the axis of the rotation meets the unit sphere (same as in complex??).
Lemma N32.4. If is a pole of and , then is a pole of .