Week 2, Sep. 21,24, 2009 1. group theory 1.2. group action.
Group action is one of the most fundamental concept in group theory.
There are many situations that group actions appear naturally. The purpose of this section is to develop basic language of group action and apply this to the study of abstract groups.
We will first define the group action and illustrate some previous known theorem as examples.
Definition 1.2.1. We say a group G acts on a set S, or S is a G- set, if there is function α : G × S → S, usually denoted α(g, x) = gx, compatible with group structure, i.e. satisfying:
(1) let e ∈ G be the idetity, then ex = x for all x ∈ S.
(2) g(hx) = (gh)x for all g, h ∈ G, x ∈ S.
By the definition, it’s clear to see that if y = gx, then x = g−1y.
Because x = ex = (g−1g)x = g−1(gx) = g−1y.
Moreover, one can see that given a group action α : G × S → S is equivalent to have a group homomorphism ˜α : G → A(S), where A(S) denote the group of bijections on S.
Exercise 1.2.2. There is a bijection between {group action of G on S}
with {group homomorphism G → A(S)}.
The group actions provide a link of the groups and the sets it acts on.
For example, one can produce many examples of groups by considering (sub)groups of bijection on certain geometric object.
Example 1.2.3.
Let S = R1. The set of all linear maps from S to S gives the group as in Example 1.1.8. The subgroup A is the group of linear transformations by regarding S as a vector space over R. ¤ Example 1.2.4.
Let S be a reaular n-gon on the plane R2. Let G be the group of rigid motions that preserves the set S. Then G ∼= Dn, the dihedral
group. ¤
Example 1.2.5.
There are five types of regular polygons: tetrahedron, hexahedron (=cube), octahedron, dodecahedron, icosahedron with 4, 6, 8, 12, 20 faces respectively. By considering their symmetry, one sees that hexahedron and octahedron have the same group of symmetry. Therefore, they are the same geometry is a sense.
1
Notice also that dodecahedron and icosahedron have the same group of geometry. In fact, this reflects the facts that they are ”dual” regular polygons by jointing the center of each faces. Tetrahedron is dual to
itself. ¤
Theorem 1.2.6 (Cayley’s Theorem). Let G be a finite group of |G| = n. Then there is an injective homomorphism G → Sn.
Sketch. To see this, we consider G acts on G via the group operation, i.e. G × G → G. Thus we have a homomorphism ϕ : G → A(G).
It’s clear that A(G) = Sn. It’s easy to see that ϕ is injective. ¤ This give an example of ”permutation representation”. That is, rep- resent a group into permutation groups. We gave an interesting exam- ple:
Example 1.2.7.
Let F2be the field of 2 elements. We would like to see that GL(2, F2) ∼= S3.
We consider V the 2 dimensional vector space over F2. There are 3 non-zero vectors in V , denoted, W := {v1 := e1, v2 := e2, v3 := e1+e2}.
It’s clear that GL(2, F2) acts on W . Thus we have a representation GL(2, F2) → A(W ) ∼= S3. One can check that this is indeed an iso-
morphism. ¤
We now introduce two important notions.
Definition 1.2.8. Suppose G acts on S. For x ∈ S, the orbit of x is defined as
Ox:= {gx|g ∈ G}.
And the stabilizer of x is defined as
Gx:= {g ∈ G|gx = x}.
It’s immediate to check the following:
Proposition 1.2.9. Given a group G acting on S. For x, y ∈ S, we have:
(1) Either Ox = Oy or Ox∩ Oy = ∅. Ox = Oy if and only if y = gx for some g ∈ G.
(2) Gx < G.
(3) If y = gx, then Gy = gGxg−1.
(4) If y = gx, then {h ∈ G|hx = y} = gGx. Proposition 1.2.10.
|G| = |Ox| · |Gx|.
Sketch. For given y ∈ Ox, we fixed an gy ∈ G such that y = gyx once and for all. Let Sy := {g ∈ G|gx = y} = gyGx is othing but the left coset. Then clearly
G = ∪y∈OxSy = ∪y∈OxgyGx, which is a disjoint union.
Furthermore, we have |Sy| = |gyGx| = |Gx|. We may define a map G → Ox× Gx as sets by g 7→ (gx, g(ggx)−1), which is a bijection. Thus
|G| = |Ox| · |Gx|.
¤ Corollary 1.2.11 (Lagrange’s Theorem). Let H < G be a subgroup.
Then |G| = |G/H| · |H|.
Proof. We take S = G/H with the action G × G/H → G/H via α(g, xH) = gxH. For H ∈ S, the stabilizer is H, and the orbit is G/H. Thus we have
|G| = |G/H| · |H|,
which is the Lagrange’s theorem. ¤
Another way of counting is to consider the decomposition of S into disjoint union of orbits. Note that if Ox = Oy if and only if y ∈ Ox. Thus for convenience, we pick a representative in each orbit and let I be a set of representatives of orbits. We have a disjoint union:
S = ∪x∈IOx. In particular,
|S| =X
x∈I
|Ox|.
This simple minded equation actually give various nice application.
We have the following natural applications.
Example 1.2.12 (translation).
Let G be a group. One can consider the action G × G → G by α(g, x) = gx. Such action is called translation. More generally, let H < G be a subgroup. Then one has translation H × G → G by (h, x) 7→ hx. In this setting, Ox = Hx. And the set of orbits is G/H, the right cosets of H in G. Then
|S| =X
x∈I
|Ox| = |G/H| · |H|
gives Lagrange theorem again. ¤
Example 1.2.13 (conjugation).
Let G be a group. One can consider the action G × G → G by α(g, x) = gxg−1. Such action is called conjugation. For a x ∈ G, Gx = C(x), the centralizer of x in G. And Ox = {gxg−1|g ∈ G} the conjugacy classes of x in G. So in general, we have
|G| = X
conj. classes
|C|,
which is the class equation.
Now assume that G is finite. The class equation now reads:
|G| =X
x∈I
|G|/|C(x)|,
where I denotes a representative of conjugacy classes.
And Ox = {x} if and only if x ∈ Z(G), the center of G. So, for G finite, the class equation now gives
|G| = |Z(G)| + X
x∈I,x6∈Z(G)
|G|/|C(x)|.
Which is the usual form of class equation. ¤ The class equation is very useful if the group is a finite p-group. We recall some definition
Definition 1.2.14. If p is a prime, then a p-group is a group in which every element has order a power of p.
By a finite p-group, we mean a group G with |G| = pn for some n > 0.
Consider now G is a finite p group acting on S. Let S0 := {x ∈ S|gx = x, ∀g ∈ G}.
Then the class equation can be written as
|S| = |S0| + X
x∈I,x6∈S0
|Ox|.
One has the following
Lemma 1.2.15. Let G be a finite p-group. Keep the notation as above, then
|S| ≡ |S0| (mod p).
Proof. If x 6∈ S0, then 1 6= |Ox| = pk because |G| = |Ox| · |Gx|. ¤ By consider the conjugation G × G → G, one sees that
Corollary 1.2.16. If G is a finite p-group, then G has non-trivial center.
By using the similar technique, one can also prove the important Cauchy’s theorem
Theorem 1.2.17 (Cauchy). Let G be a finite group such that p | |G|.
Then there is an element in G of order p.
sketch. We keep the notation as in Lemma 1.2.15. Let S := {(a1, ..., ap)|ai ∈ G,Y
ai = e}.
And consider a group action Zp×S → S by (1, (a1, .., ap)) 7→ (ap, a1, ..., ap−1).
One claims that S0 = {(a, a, ..., a)|a ∈ G, ap = e}.
By the Lemma, one has |S| ≡ |S0| (mod p). It follows that p | |S0|.
In particular, |S0| > 1, hence there is (a, ..., a) ∈ S0 with a 6= e. One
sees that o(a) = p. ¤
Corollary 1.2.18. A finite group G is a p-group if and only it is a finite p-group.
