Chapter 4 Definitions

Question 1

Group Action

Answer 1

Let G be a group and let Ω be a set. Then G acts on Ω if for all g ∈ G and for all α ∈ Ω. There exists g . α ∈ Ω
such that (1) for all α ∈ Ω we have e . α = α
and
(2) for all g,h ∈ G and for all α ∈ Ω we have h . (g . Ω ) = (hg) . α

• Ω is called the G- set

Question 2

Regular Action

Answer 2

Let G be any group and let the set Ω be G.

Then define the action of G on Ω by g . x = gx

Question 3

Lemma 4.5 - group that acts on set Ω , the map σ_{g}

Answer 3

Let G be a group that acts on the set Ω for g ∈ G we define the map σ_{g} : Ω → Ω by σ_{g} ( α ) = g . α
then σ_{g} ∈ Perm (Ω)

*The action of every element of the group gives a 1-1 and onto map on Ω. when |Ω| = n < ∞ then the action of every element of G gives an element of S_{n}

Question 4

Translation Action (Action of H on G by left multiplication )

Answer 4

Let G be any group. let H be a subgroup of G then define an action of H on Ω = G by h . g = hg

Question 5

Conjugate Action

Answer 5

Let G be any group and let Ω =G. For g ∈ G and x ∈ Ω define g . x = g x g^{-1}

Question 6

Another Conjugate Action

Answer 6

Let G be any group and let Ω be the set of subgroups of G. Define an action of G on Ω by g . H = g H g^{-1} for g ∈ G and H is subgroup of G