Sep. 29, 2006 (Fri.) 2.3. 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 2.3.1. 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 2.3.2. There is a bijection between {group action of G on S}
with {group homomorphism G → A(S)}.
Example 2.3.3 (Cayley’s Theorem).
Let G be a finite group of |G| = n. Then there is an injective homo- morphism G → Sn.
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 another example:
Example 2.3.4.
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 vector 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 2.3.5. 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:
Lemma 2.3.6. Given a group G acting on S. For x, y ∈ S, we have:
1. Gx < G.
2. either Ox = Oy or Ox∩ Oy = ∅.
3. if y = gx, then Gy = gGxg−1. Proposition 2.3.7.
|G| = |Ox| · |Gx|.
Sketch. For given y ∈ Ox, we consider Sy := {g ∈ G|gx = y}. Then G = ∪y∈OxSy,
which is a disjoint union.
Furthermore, for each y ∈ Ox, we can write y = gx. Then one has Sy = gGx. In particular |Sy| = |Gx|. We fix a gy such that y = gyx once and for all. We may define a bijection G → Ox× Gx as sets by g 7→ (gx, g(ggx)−1). Thus
|G| = |Ox| · |Gx|.
¤ Corollary 2.3.8 (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 2.3.9 (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 2.3.10 (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 2.3.11. 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 2.3.12. 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 2.3.13. 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 2.3.14 (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 2.3.12. 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 2.3.15. A finite group G is a p-group if and only it is a finite p-group.
2.4. Sylow’s theorems. We are now ready to prove Sylow theorems.
The first theorem regards the existence of p-subgroups in a given group.
The second theorem deals with relation between p-subgroups. In par- ticular, all Sylow p-subgroups are conjugate. The third theorem counts the number of Sylow p-subgroups.
Theorem 2.4.1 (First Sylow theorem). Let G be a finite group of order pnm (where (p, m) = 1). Then there are subgroups of order pi for all 0 ≤ i ≤ n.
Furthermore, for each subgroup Hi of order pi, there is a subgroup Hi+1 of order pi+1 such that HiC Hi+1 for 0 ≤ i ≤ n − 1.
In particular, there exists a subgroup of order pn, which is maximal possible, called Sylow p-subgroup. We recall the useful lemma which will be used frequently.
Lemma 2.4.2. Let G be a finite p-group. Then
|S| ≡ |S0| (mod p).
proof of the theorem. We will find subgroup of order pi inductively. By Cauchy’s theorem, there is a subgroup of order p. Suppose that H is a subgroup of order pi. Consider the group action that H acts on
S = G/H by translation, i.e. H × G/H → G/H by h(xH) := hxH.
One shows that xH ∈ S0 if and only if xH = hxH for all h ∈ H if and only if x ∈ NG(H). Thus |S0| = |NG(H)/H|.
If i < n, then
|S0| ∼= |S| = pn−im ≡ 0 (mod p).
By Cauchy’s theorem, the group NG(H)/H contains a subgroup of or- der p. The subgroup is of the form H1/H, hence |H1| = pi+1. Moreover,
H C H1. ¤
Example 2.4.3. If G is a finite p-group of order pn, then one has a series of subgroups {e} = H0 < H1 < ... < Hn = G such that |Hi| = pi and HiC Hi+1, Hi+1/Hi ∼= Zp. In particular, G is solvable.
Definition 2.4.4. A subgroup P of G is a Sylow p-subgroup if P is a maximal p-subgroup of G.
If G is finite of order pnm then a subgroup P is a Sylow p-subgroup if and only if |P | = pn by the proof of the first theorem.
Theorem 2.4.5 (Second Sylow theorem). Let G be a finite group of order pnm. If H is a p-subgroup of G, and P is any Sylow p-subgroup of G, then there exists x ∈ G such that xHx−1 < P .
Proof. Let S = G/P be the set of left cosets and H acts on S by translation. Thus by Lemma 2.3.12, one has |S0| ≡ |S| = m(mod p).
Therefore, S0 6= ∅. One has
xP ∈ S0 ⇔ hxP = xP ∀h ∈ H ⇔ x−1Hx < P.
This completes the proof. ¤
An immedaitely but important consequence is that any two Sylow p-subgroups are conjugate.
Theorem 2.4.6 (Third Sylow theorem). Let G be a finite group of order pnm. The number of Sylow p-subgroups divides |G| and is of the form kp + 1.
Proof. Let S be the conjugate class of a Sylow p-subgroup P (this is the same as the set of all Sylow p-subgroups). We consider the action that G acts on S by conjugation, then the action is transitive, i.e. for any x, y ∈ S, there exists g ∈ G such that y = gx. In particular Ox = S.
Hence |S| | |G| for |G| = |Gx| · |Ox|.
Furthermore, we consider the action P × S → S by conjugation.
Then
Q ∈ S0 ⇔ xQx−1 = Q ∀x ∈ P ⇔ P < NG(Q).
Both P, Q are Sylow p-subgroup of NG(Q) and therefore conjugate in NG(Q). However, Q C NG(Q), Q has no conjugate other than itself.
Thus one concludes that P = Q. In particular, S0 = {P }. By Lemma
2.3.12, one has |S| = 1 + kp. ¤
Example 2.4.7.
Group of order 200 must have normal Sylow subgroups. Hence it’s not simple. To see this, let rp := number of Sylow p-subgroups. Then r5 = 1. So if P is a Sylow 5-subgroup. Since gP g−1 is also a Sylow subgroup, it follows that gP g−1 = P for all g ∈ G. Thus P C G. ¤ Example 2.4.8.
There is no simple group of order 36. To see this, we consider P a Sylow 3-subgroup. Then r3 = 1 or 4. In case that r3 = 4, let S be the set of Sylow 3-subgroups. We have a group action G × S → S by conjugation. Thus we have a group homomorphism ϕ : G → A(S) ∼= S4. Comparing the cardinality of groups, one sees that ϕ must have non-trivial kernel. Hence G is not simple. ¤ 2.5. groups of small order. We can use the technique developed in the previous sections to study group of small order in more detail.
First of all, as a direct consequence of Cauchy’s theorem,
Proposition 2.5.1. Let p be a prime. A group of order p is cyclic.
Example 2.5.2.
Classify groups of order 2p.
If p = 2, then this is well-known. So we may assume that p > 2.
First of all there is a subgroup H < G of order p, generated by x, by Cauchy’s theorem. By Sylow’s third theorem, we have rp = 1, hence H is normal. Similarly, there is an element of order 2, say y. By normality of H, we have yxy−1 = xk for some k. Since
x = y2xy−2 = yxky−1 = xk2, it follows that k2 ≡ 1( mod p). Hence k ≡ 1 or ≡ −1.
Case 1. k ≡ 1, then xy = yx. It follows that G is abelian. By chinese Remainder Theorem, G is cyclic.
Case 2. k ≡ −1, then xy = yx−1. These kind of group is called
dihedral groups, denoted D2p. ¤
Example 2.5.3.
Let p, q be primes. If |G| = pq, then its structure can be determined similarly.
We assume that p > q. Then there are x, y ∈ G of order p, q respec- tively. Moreover, H :=< x > CG. We have yxy−1 = xk for some k.
Since
x = yqxy−q = yxky−1 = xkq,
it follows that kq ≡ 1( mod p). Now the situation depends on the structure of Z∗p. Recall that Z∗p ∼= Zp−1 is cyclic.
Case 1. q - p − 1, then kq ≡ 1( mod p) implies that k ≡ 1. Hence xy = yx. It follows that G is abelian. By chinese Remainder Theorem,
G is cyclic.
Case 2. q | p − 1, then kq ≡ 1( mod p) has q solutions, k ≡ a, a2, ..., aq−1, aq≡ 1. If we pick k ≡ a, then we determined a group G1 which is generated by x1, y1 with y1x1y1−1 = xa1. If we pick k ≡ a2, then we determined a group G2 which is generated by x2, y2 with y2x2y2−1 = xa22. Note that the map ϕ : G2 → G1 by ϕ(y2) = y12, ϕ(x2) = x1 gives an isomorphism. Therefore, for different solution k ≡ a, a2, ..., aq−1,
they determined the same group. ¤
There is a useful construction to produce groups from simple ones called semi-direct product which we now introduce. Given two groups G, H and a homomorphism θ : H → Aut(G). Let G×θH be the set G × H with the binary operation (g, h)(g0, h0) = (g(θ(h)(g0)), hh0).
One can verify that this produce a group.
For example, in the case 2 of above example, we have G = Zp, H = Zq and we consider θ : Zq → Aut(Zp) ∼= Z∗pby θ(1) = a. Then we obtained Zp ×θ Zq. Such group is called a metacyclic groups.
Proposition 2.5.4. Let p be a primes. If |G| = p2, then G is abelian.
We will discuss the structure of finite ableina groups later. In prin- ciple, their structure are pretty easy.
sketch. By class equation, one sees that Z(G) is non-trivial.
Case 1. if |Z(G)| = p2, then G is abelian.
Case 2. if |Z(G)| = p, then G/Z(G) is a group of order p , hence cyclic. We pick x ∈ G such that G/Z(G) is generated by xZ(G). We also pick y ∈ G such that Z(G) is generated by y. It’s easy to check that G is generated by x, y. Note that xy = yx, it follows that G is
abelian. ¤
Using above properties, one can classified groups of order ≤ 15 com- pletely unless for order 8 and 12. In fact groups of order 8 are either abelian or D8 or Q8. Where Q8 is the quaterion group defined by {i, j, k, −i, −j, −k, 1, −1|i2 = j2 = k2 = −1, ijk = −1}.
Easy example of non-abelian groups of order 12 includes A4, D12. In fact there is one more, T =< a, b|a6 = b4 = 1, b2 = a3 = (ab)2 >.
Theorem 2.5.5. every non-abelian group G of order 12 is isomorphic to A4, D12 or T .
sketch. Let P be a Sylow 3-subgroup. We first consider the action G × G/P → G/P by translation. It gives rise to a homomorphism ϕ : G → A(G/P ) ∼= S4. It’s clear that ker(ϕ) < P .
Case 1. ker(ϕ) = {e}, then G ∼= A4.
Case 2. ker(ϕ) = P . Then we need to wok harder. So now, P C G and P is the unique Sylow 3-subgroup. Let P = {x, x2, x3 = e}, then x, x2 are the only element in G of order 3.
Let K be a Sylow 2-subgroup, then K is either V4 or Z4.
Case 2.i. If K ∼= V4, by computing the relation between generators,
one can show that G ∼= D12.
Case 2.ii. If K ∼= Z4, by computing the relation between generators, one can show that G ∼= T .
¤ Groups of order pn, n ≥ 3 could be very complicated. Here just give two more examples.
Example 2.5.6.
Let G < GL(2, C) be the group generated by A =
µ 0 ω ω 0
¶ and B =
µ 0 1
−1 0
¶
, where ω is a primitive 2n−1th root of unity for n ≥ 3.
Then G is a group of order 2n. ¤
Example 2.5.7.
Let G < GL(3, C) be the group generated by A =
1 0 0
0 ω 0
0 0 ω2
and B =
0 1 0 0 0 1 1 0 0
, where ω is a primitive 3th root of unity. Then
G is a group of order 27. ¤
