S2, C5: Group Actions

Question 1

A group G acts on a non-empty set X if, for each g ϵ G and each x ϵ X, there is an element g*x ϵ X s.t….

Answer 1

GA1: ex = x for all xϵX.
GA2: g(h*x) = (gh) * x for all g, h ϵ G and all x ϵ X.

Question 2

Let G be a group acting on a non-empty set X.

For any x∈X, define the orbit of x?

Answer 2

orb(x) = {y ∈ X : y = g ∗ x for some g ∈ G}

Question 3

Let G be a group acting on a non-empty set X.

For any x∈X, define the stabiliser of x?

Answer 3

stab(x) = {g ∈ G : g ∗ x = x}

Question 4

What is the alternating polynomial?

Answer 4

For n > 2, an is the product of all polynomials of the form xi - xj with i

Question 5

Let G be a group acting on a non-empty set X.

For each y∈orb(x), the sending set sendx(y) is given by

Answer 5

sendx(y) = {g∈G : g*x=y}

This consists of those elements of G that send x to y. Notice that sendx(x) = stab(x).

Question 6

Let G be a group acting on a non-empty set X.

For each g∈G, the fixed set of g is the subset

Answer 6

fix(g) = {x ∈ X : g*x = x}

This consists of those elements that are fixed by g.

Question 7

The Orbit Counting Theorem. Let G be a finite group acting on a non-empty set X and let n be the number of orbits. Then n=

Answer 7

n = 1 / |G| * (sum for all g∈G) |fix(g)|