Download Actions of Groups on Sets

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 Gaction.
— 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. If H ≤ G then define a right action of H on G by right multiplication: gh.
Example 2. Let G = GLn (R) and X = Rn , with a left action defined by multiplying a
vector by a matrix: A x. (Here R is a commutative ring with unitsy, e.g., Z, R, C, Q, etc.)
Example 3. Let G = Z and X = R, with a right action defined by ordinary addition: x + n.
Example 4. 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 5. G = GL2 (R) acts from the left on the upper half plane H = {z = x+iy | y > 0}
by linear fractional transformations (sometimes called fractional linear transformations):
az + b
a b
z=
.
(1)
c d
cz + d
Example 6. Let X = W × · · · × W , the n-fold Cartesian product of the set W , and let
G = Σn , the symmetric group on n letters. Then G acts on X from the left by permuting
the components of elements of X.
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Remark. 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 7. Let O(n) ≤ GLn (R) be the subgroup of n×n orthogonal matrices and X = Rn ,
with a left action defined as in Example 2 above. Then restriction gives us a left action of
O(n) on X. (If R = C in GLn (R) we use U (n), the n × n Hermitian matrices.)
Example 8. SL2 (Z) = {A ∈ GL2 (R) | det(A) = 1 and A has integer entries} ≤ GL2 (R)
acts on H as in Example 5 above by restriction.
Remark. 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 9. Let Y = B n ⊂ X = F n be the closed unit ball and O(n) act from the left on
X as in Example 7 above. Since multiplication by orthogonal matrices preserves the length
of a vector, Y is invariant under the action. We thus obtain a left action of O(n) on Y . In
fact, we also get an action of O(n) on S n−1 , the unit sphere in F n . (Again, we use U (n) in
the case where F = C.)
Example 10. Z2 ' {±I } ≤ GL3 (R) acts on S 2 = {x ∈ R3 | kxk = 1} by a restriction of
the group to a subgroup and of the set to an invariant subset. (See Example 2.) If we think
of Z2 as generated multiplicatively by T , then the action is simply
T x = −x.
(2)
Example 11. 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 the function
Φ : G → SX defined by Φ(g) = Φg
(3)
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.
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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 12. In Example 2 above for R = R or R = C, the orbit set is the two-point set
{[0], [v]}, where v is any non-zero vector in Rn (respectively Cn ). The topology of the orbit
space is {∅, [v], {[0], [v]}}—why?—which is not Hausdorff.
Example 13. In Example 3 above, the orbit space is S 1 , the unit circle in R2 with the
standard topology.
Example 14. In Example 10 above, the orbit space is RP 2 , the real projective plane.
Example 15. The orbit set of the product action defined in Example 11 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 3, then X = R2 and X 0
is the torus S 1 × S 1 .
Example 16. In Example 4 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, 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. (Check.)
Definitions.
(i) If Gx = e for all x ∈ X we call the action free.
(ii) If Gx = G for all x ∈ X we call the action trivial.
(iii) If ∩x∈G Gx = {e} we call the action faithful (or effective). This is equivalent to saying
that if gx = x for all x ∈ X then g = e.
(iv) 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.
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Remarks. (i) A free action is fixed point free, but the converse is not necessarily true. The
reader should supply a counterexample.
(ii) An action is faithful if and only if ker Φ = {e} in (3) above. If an action is not faithful
and N = ker Φ, then G/N acts on X by (gN )x = gx, and this action is faithful.
Example 17. SL2 (Z) does not act faithfully on H in (8) above; ker Φ = {±I }. So P SL2 (Z) =
SL2 (Z)/{±I } acts faithfully on H. P SL2 (Z) is a very important group, called the modular
group.
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 1 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:
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.
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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 2016
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