I.3 Group Actions and I.4 the Sylow Theorems

Definition 1.3.1

An Action G on X is a group map GxX \rightarrow satisfying 1) 1x=x \forall x\in X 2) g*h(x)=(gh)(x) \forall g,h\in G and \forall x\in X

Lemma 1.3.2

Let X is a g-set and let g,h\in G and x,y\in X. Then a) If g*x=g*y then x=y and b) g*x=h*y=> (h^{-1}g)*x=y c) the map \sigma_g : X \rightarrow X is bijective with the inverse \sigma_{-g}

Definition 1.3.3

For any set X\ne \emptyset, S_x:={\alpha: X\rightarrow X| \alpha is bijective} is a group with respect to compositions as multiplication called the group of permutations of X.

Renark 1.3.4

If |X|=|Y|, then the group of permutations is isomorphic. The proof is in the Homeworks

Proposition 1.3.5

If X is a G set and \sigma_g:X\rightarrow X as defined in 1.3.2c, then we can define \sigma:G\rightarrow S_X by \omega(g)=\omega_g, \forall g\in G is a group homomorphism. If a group homomorphism \varphi: G\rightarrow S_X is given, then X becomes a G set via g*x=\varphi(g)(x) \forall g\in G, \forall x \in X.

Remark 1.3.6

If X is a G-set and \sigma: G\righarrow S_X as in 1.3.5.a , then we have that \ker(\sigma)=\{g\in G| \sigma(g)=id_x\}

Definition 1.3.7

The action of G on X is called faithful if \ker(\sigma)=\{1\} and is called transitive if for any given x,y\in X, \exists g\in G such that g*x=y

Example 1.3.8

a. The trivial action of G on X is defined by g*x=x \forall g\in G \forall x\in X. Observe that ker(\sigma)=G and thus \sigma(G)=\id(x)

b. F is a field then the group GL_n(F)=\{A\in M_n(F)| det(A)\ne 0\} acts on X=F^n by multiplication of matrices: A*x=Ax. We can do this forall A \in GL_n(X) and \forall x\in X. This action is faithful and transitive'

c) Any G\le S acts on I_n by circulation. For example, for g\in G, we have that g*i=g(i). This action is always faithful since if g*I=I, \forall I\in I_n implies that g=id

d) Z_n^x acts on Z_n by multiplication where \bar{a}*\bar{b}=\bar{ab}. Its not transitive but it is faithful

e) G acts on X=G by left multiplication: g*x=gx. This action is faithful and transitive

f) Given H\le G, X=G/H is a factor group if the action of G on H by left multiplication: X=gH. This action is transitive not faithful. Let \sigma: G\rightarrow S_X

Proposition 1.3.9

a) Every group G is isomorphic to subgroup of S_G

b) If G is finite, |G|=n, then G is isomorphic to a subgroup of S_n

Definition 1.3.10

Let X be a G-Set and x\in X. Then we have the following:

a) S_G(X):=\{g\in G| g*X=X\} is the stabilizer of x in G. If G is understood, we also say S(X)=S_G(X).

b) G*x=\{g*x| g\in G\} \subseteq X is called the orbit of X under action of G

Lemma 1.3.11

S(x) \le G and |G:S(x)|=|G*x|

Example 1.3.12

The action of G on itself by conjugation. The orbit of x is G*x = \{gxg^{-1}|g\in G\}:= class of x = conjugacy class of x\in G

Application 1.3.13

For G=S_n, determine C_{S_n}(\sigma) if \sigma \in S_n is an n-cycle. We have that always <X>\le C_G(X). Therefore, we have that <\sigma>\le C_{S_n}(\sigma) and we have that |<\sigma>|=O(\sigma)=n. Thus the class \sigma = \{ all n cycles of S_n \}.

Definition 1.3.14

Let G acts on X=G with conjugation. Let the group homomorphism \sigma: G\rightarrow S_G where g \mapsto \sigma_g and we have that \sigma_g: G\rightarrow G where x\mapsto gxg^{-1}. \sigma is a group homomorphism and thus an isomorphism. We call it an automorphism. Define Aut(G)=\{ \alpha:G \rightarrow G| \alpha is an automorphism of G \} is a group with respect to composition. Furthermore, we have that Aut(G)\le S_G.

Lemma 1.3.15

If X is a G set then

a) S(g*x)-gS(x)g^{-1}

b) If the action of g on S is transitive, and \sigma: G\rightarrow S_x is usual then \ker \sigma = \Bigcap g\in G gHg^{-1} where H=S(x) for some x\in X.

Corollary 1.3.16

If H \le G and |G:H|=n\in N, then \exists a group homomorphism \varphi: G\rightarrow S_n and \ker{\varphi} = \bigcup_{g\in G} gHg^{-1}. \varphi is called the permutation representation

Application 1.3.17

A_5 has no subgroup of order 15.

Proposition 1.3.18

Let G be a finite group with |G|=p_1^\delta_1,…p_m^{\delta_m} with primes p_1<…<p_m and \delta_i\in N \forall i< m. if H \le satisfies that $|G:H|=p_1| then H\triangleleft G.

Proposition 1.3.19

Let X be a G Set. Then:
a) If x\in X and y\in G\cdot x, then G \cdot y = G \cdot x
b) For any x,y \in X, either G \cdot x = G \cdot y or their intersection is empty.
c) If one picks elements x_i in each G orbit in X, then X is the disjoint union of the orbits, G \cdot x_i

Corollary 1.3.20

If X is a finite G set and if we pick one element from 1\le I \le m in each m disjoint G orbits of x then |X|=\Sigma_{i=1}^{m}|G\cdot x_i| = \Sigma_{i=1}^{m} |G:S(x_i)|

Theorem 1.3.21

If G is a group, x\in G, then |class x|=1 if and only if x\in Z(G) and has representative x_i for the m conjugacy class that we have at least 2 elements |G|=|Z(G)|+\Sigma^m_{I=1}|class (x_i)|= |Z(G)|+\Sigma_{i=1}^{m}|G:C_G(x_i)|

Definition 1.3.22

If p is prime, a finite group G is called a p group if |G|=p^k for some k\in N_0

Ex: D_4 and Q are 2 groups of order 8

Proposition 1.3.23

If G is a nontrivial p group, then Z(G)\ne {1}

Corollary 1.3.24

If |G|=p² for some prime p, then G is abelian

Example 1.3.25

The class equations on S_4 and A_4
G=S_4, |G|=24, Z(G)={1}. Thus we get that 24=1+\Sigma^m_{i=1} class x_i with x_i\in C_i for nontrivial classes. These classes correspond to the cycle types:
1) id =1
2) products of 2 cycles = 3
3) 3 cycles = 8
4) 2 cycles = 6
5) 4 cycles = 6
24=1+3+8+6+6

Application for Centralizers: C_{S_4}(123)=|S_4|/class(123)=3

A_4 class equation: |A_4|=12=1+3+8 cannot be true since 8 does not divide 12, so It cannot appear in the class equation. The set of all three cycles split into 2 conjugacy classes that are not all conjugate. Thus the correct class equation would be 12=1+3+4+4