Download Group Actions

Group Actions
Definition. Given a group G with identity element e and a set X, define a right action of G on X to be a
function
X ×G→X
(x, g) 7→ xg
satisfying
(i) (xg)h = x(gh) ∀ h, g ∈ G, x ∈ X
(ii) xe = x ∀ x ∈ X
We call X a right G-set.
Remarks. We define left G-actions and left G-sets in an analogous manner.
Any right G-action can be made into a left G-action by defining gx = xg −1 .
If the group G is abelian, we traditionally write the group operation and the action additively. For example,
if G = (Z, +), we write x + n instead of xn for a right G-action.
In topology we often want X to be a topological space, G to be a topological group (although possibly with
only the discrete topology), and the function defining the action to be continuous. We then refer to X as a
G-space—right or left.
Example 1. Let G = GLn (F ) and X = F n , with a left action defined by multiplying a vector by a matrix:
A x. (Here F is a field, e.g., R, C, Q, etc.)
Example 2. Let G = Z and X = R, with a right action defined by ordinary addition: x + n.
Example 3. Let G = S 1 , the multiplicative group of complex numbers of norm 1, and let X = C, the
complex plane. Define a right action by complex multiplication: wz.
Example 4. For any action—right or left—of G on X, if H is a subgroup of G, then H acts on X by
restriction.
Example 5. For any action—right or left—of G on X, if Y is a subset of X such that yg ∈ Y ∀g ∈ G
(respectively gy ∈ Y ∀g ∈ G), then G acts on Y by restriction. We say that such a Y is invariant under the
action of G or is G-invariant.
Example 6. If Gi acts from the right on Xi for i = 1, . . . n, then G1 × . . . × Gn acts from the right on
X1 × . . . × Xn by (x1 , . . . , xn )(g1 , . . . , gn ) = (x1 g1 , . . . , xn gn ). We can define a left action of the product
group on the product set similarly.
Definitions. As usual, a permutation of X is a one-one onto function from X to itself. SX , the set of all
permutations or symmetries of X, is a group under composition of functions.
Given a left G-action on X, and given a fixed element g ∈ G, define Φg : X → X by Φg (x) = gx.
Remarks. Φg ∈ SX ∀ g ∈ G, and Φ : G → SX defined by Φ(g) = Φg is a group homomorphism. (A similar
statement holds for right G-actions, but we must define our group of symmetries of X as acting from the
right as well, i.e., we must write functional values as (x)f and compositions as (x)(f ◦ g) = ((x)f )g.)
If X is a topological space and the action is continuous then Φg ∈ Aut(X), the group of all self-homeomorphisms—
automorphisms—of X. Moreover, in that case Φ : G → Aut(X) is a homomorphism.
1
Definitions. For a given x ∈ X, xG = {xg : g ∈ G} is called the orbit of x. The orbits partition X, i.e.,
either xG = yG or xG ∩ yG = ∅. Equivalently, x ∼ y if x = yg for some g ∈ G is an equivalence relation on
X. The orbit set X 0 of X is X/ ∼, the set of the equivalence classes under ∼. It may be that there is only
one equivalence class—X = xG. We say in that case that the action is transitive: we can get from any x in
X to any y by translation, i.e., y = xg for some g ∈ G.
Remark. If X is a topological space, we give X 0 the quotient topology and call it the orbit space of X.
Example 7. In Example 1 above, the orbit set is the two-point set {[0], [v]}, where v is any non-zero vector
in F n . The topology of the orbit space is {∅, [v], {[0], [v]}}—why?—which is not Hausdorff.
Example 8. In Example 2 above, the orbit space is S 1 , the unit circle in R2 with the standard topology.
Example 9. The orbit set of the product action defined in Example 6 above is just the product of the
individual orbit sets, i.e., (X1 × . . . × Xn )0 = X10 × . . . × Xn0 . In particular, if we have n = 2 and we take two
copies of the action from Example 2, the orbit space is the torus S 1 × S 1 .
Example 10. In Example 3 above, the orbit space is the ray [0, ∞), the non-negative x-axis in R2 with the
standard topology.
Definition. Gx = {g ∈ G : xg = x} is called the isotropy subgroup of x (or also, the stabilizer of x).
Remark. Gx is a subgroup of G. Hence if G is finite, then the order of Gx divides the order of G.
How are Gx and Gy related for x 6= y? At least if y = xg we have Gy = g −1 Gx g.
Definitions. If Gx = e ∀x ∈ X we call the action free.
If Gx = G ∀x ∈ X we call the action trivial.
If Gx = G we call x a fixed point of the action. An action is called fixed point free if it has no fixed points.
Remark. A free action is fixed point free, but the converse is not necessarily true. The reader should supply
a counterexample.
Definition. If X and Y are two left G-sets—same G—and if f : X → Y is a function, we call f equivariant
if f (gx) = gf (x) ∀g ∈ G, x ∈ X. Similarly for right G-sets. If X and Y are G-spaces, we require f to be
continuous.
Remark. An equivariant function f : X → Y induces a function between the orbit sets: f 0 : X 0 → Y 0 , defined
by f 0 ([x]) = [f (x)]. This function f 0 is continuous if f is continuous and the orbit spaces have the quotient
topology. (Let the reader verify these claims.)
Suppose G is a topological group and H ≤ G is a subgroup. By Example 5 we always have a right action
of H on G:
G×H →G
(g, h) 7→ gh
If H = G this action is transitive, since for g1 and g2 , g2 = g1 (g1−1 g2 ). On the other hand, if H G is
a proper subgroup, we have an action of H on G that is no longer transitive. In fact, the orbit of g under
this action is just the left coset gH. (Note that the orbits of this right action are left cosets.) If G is finite,
Lagrange’s Theorem then implies that the cardinality of the orbit of g must divide the order of G. This is
in fact true for arbitrary G-sets X:
2
Let xG be the orbit of an element x in the G-set X. Consider the onto function
φ : G → xG
g 7→ xg.
Let g1 ∼ g2 if φ(g1 ) = φ(g2 ). As usual, this is an equivalence relation, so we get an induced map φ̂ defined
by φ̂([g]) = φ(g). This φ̂ is one-one. But note that φ(g1 ) = φ(g2 ) implies xg1 = xg2 , xg1 g2−1 = x, and hence
g1 g2−1 ∈ Gx , the isotropy subgroup of x. It follows that the equivalence classes of ∼ are exactly the right
cosets Gx g of Gx in G, i.e., G/ ∼ = G\Gx .
Now G\Gx has a natural right G-action: (Gx g1 )g = Gx (g1 g), and with respect to this action, φ̂ is
equivariant: φ̂((Gx g1 )g) = φ̂(Gx (g1 g)) = x(g1 g) = (xg1 )g = φ̂(Gx g1 )g. So we have a one-one, onto,
equivariant function
φ̂ : G\Gx → xG.
This is an equivalence or isomorphism in the category of G-sets.
Remarks. If G is finite, then |G\Gx | = (G : Gx ) divides |G|, so |xG| divides |G| as claimed.
If G is a finite group, then φ̂ is also an isomorphism in the category of G-spaces, since G\Gx and xG will
both have the discrete topology.
If G is not finite, φ̂ will be one-one, onto, continuous and equivariant, but its inverse needn’t be continuous.
So φ̂ needn’t be a G-isomorphism, i.e., an equivariant homeomorphism. However, if X is Hausdorff and G
is compact, then φ̂ will be a G-isomorphism.
c R. Kubelka 2012
3