Transformation Geometry — Math 331

April 26, 2004

Transformation Groups: III

Definition. If G is a group of transformations of a set X, g an element of G, S a subset of X, and g(S) the set to which S is carried by g, one says that g stabilizes S if g(S) = S. The set of all transformations g in G that stabilize S is called the stabilizer of S in G. A subset F of G stabilizes S if it is contained in the stabilizer of S in G, i.e., if g(S) = S for every g in F.

Example. If γ is a glide reflection of the plane R^{2}, l its axis, and G the set of isometries of the form γ^{k} for k = 0, ±1, ±2, …, then G is a group that stabilizes l and is a proper subgroup of the stabilizer of l in the group of all isometries of R^{2}.

Proposition 1. For any subset S of X the stabilizer of S in G is a subgroup of G.

Proposition 2. If G is a group of transformations of X, S a subset of X, g an element of G, T = g(S), and H the stabilizer of S in G, then the stabilizer of T in G is the conjugate of H by g, i.e., the subgroup

gHg^{−1} = {ghg^{−1} | h ∈ H} .

Definition. If G is a group and H a subgroup of G, H is normal in G if gHg^{−1} = H for every g in G.

Definition. If G is a group of transformations of a set X and S a subset of X that is stabilized by G, one says that G is transitive on S if for each pair x, y of points of S there is at least one element g in G such that y = g(x). G is simply transitive on S if for each pair x, y of points of S there is exactly one element g in G such that y = g(x).

Example. The isotropy group at the origin in the group of isometries of R^{2} stabilizes the unit circle and is transitive on the unit circle but not simple transitive on the unit circle.

Example. The group of translations of R^{n} is simply transitive on R^{n}.

Assignment for Wednesday, April 28

1. If G is a group of transformations of a set X and S is a subset of X that is stabilized by G, does it then follow that G is a group of transformations of S?

2. What subsets of R^{3} are stabilized by the isotropy group of the Euclidean group at the origin?

3. What is the stabilizer of the first coordinate axis in the group of all isometries of R^{2}?

4. Prove Propostion 1.

5. Prove Propostion 2.

6. Let G be a group of transformations of a set X with G transitive on X, and let x_{0} be a given point of X. Prove that if the isotropy group of G at x_{0} is a normal subgroup of G, then it consists of only the identity transformation of X.

7. Show that the group of translations of R^{n} is normal in both the isometry group of degree n and the affine group of degree n, but that the isometry group of degree n is not normal in the affine group of degree n.