Chapter 5 GROUP ACTIONS

Question 1

what does it mean if G acts on Ω

Answer 1

for each g in G and for each alpha in Ω, there exists a unique elements αg such that

(A1) g1,g2 in G, α in Ω, then α(g1g2)=(αg1)g2
(A2) for all α in Ω, α1=α

Question 2

Suppose Ω is a G-Set α^G =….

Answer 2

α^G = {αg | g in G}

Question 3

If Ω is a G-set, then Ω is the ….

Answer 3

disjoint union of its G-orbits

PROOF: Equivalence relation between alpha and beta

Question 4

What is the stabilizer of α? (in G)

Answer 4

Gα={g ∈ G | αg=α}

Question 5

If G is a group and Ω is a G-set, then what is the relation between the stabilizer Gα and G.

Answer 5

Gα≤G

Proved using the subgroup criterion and the action axioms

Question 6

If G is a finite group and Ω is a finite G-set then i), ii)

Answer 6

i) |Δ| = [G:Gα] = |G| / |Gα|
ii) α,β in omega, where αg=β, some g in G, [note α^G=β^G]

Then Gα^g =Gβ

Proof i) show alpha g = alpha gi = Δ then show no duplicated by saying gi=gj

Proof ii) Gα^g =Gβ by subsets both ways.

Question 7

If Δ is a G-orbit, then for any α,β∈Δ, Gα and Gβ are

Answer 7

conjugate in G

Question 8

If G is a finite group and Ω is a finite G-set, then Ω is

Answer 8

the disjoint union of its G-orbits. Δ1UΔ2U…UΔm

Δi=/=Δj=/= 0 if i=/=j

Question 9

If G is a finite group and Ω is a finite G-set,

Answer 9

Gα≤G

Question 10

|Ω|=

Answer 10

sum of mod Δi = sum of [G:Gαi]

Question 11

Cauchy’s Theorem.

Answer 11

If G a finite group, p a prime number. If p | |G| then G has at least one element of order p.

Question 12

What does it mean to say G acts transitively on omega

Answer 12

If G acts upon a set Ω and Ω is a G-orbit, then we say acts transitively on Ω.

Question 13

Burnsides theorem

Answer 13

G and omega finite, G has t orbits in omega.

t=1/|G| sum |fixΩ(g)|

where fix(g) = {α in omega | αg=α}