Elementary Abstract Algebra- Examples and Applications, 2019a

19.2 SYMMETRIES OF REGULAR POLYHEDRA 781
Finally, we must show that the map φ is onto: that is, every coset of G x
is in the range of φ. This is much quicker than the proof of 1-1. Let gG x be
any left coset. If gx = y, thenφ(y) =gG x . Thus gG x is in the range of φ,
and the proof is finished.
□
At this point, it is straightforward to put Proposition 19.2.20 together
with Lagrange’s Theorem to obtain:
Proposition 19.2.21.(Orbit-Stabilizer Theorem): Let G be a group and
X a G-set. Given x ∈ X, letO x be the orbit of x under G, andletG x be
the stabilizer subgroup for the element x. then|G| = |O x |||G x |.
Exercise 19.2.22. Prove Proposition 19.2.21.
♦
The Orbit-Stabilizer Theorem enables us to quickly find some nifty relationships
among numbers of faces, vertices, and edges:
Exercise 19.2.23.
(a) Show using Proposition 19.2.21 that for the cube, the ratio (number of
edges / (number of faces) = 4/2.
(b) Use the same method to find the ratio (number of edges) / (number of
vertices) for the cube.
(c) For the dodecahedron (regular polyhedron with 12-sided faces), find the
ratio of (number of faces) / (number of vertices). (*Hint*)
♦
19.2.4 Representing a symmetry group in terms of stabilizer
subgroups
We can approach the structure of the group of rotational symmetries of a
cube from another direction. We’ve talked about stabilizer subgroups, and
we can see how these subgroups “fit together” within G. For example, we’ve
seen that for every face there are three rotations (besides the identity) that
leaves that face fixed. These rotations correspond to 90, 180, and 270 degree