10. Group Actions

Question 1

Matrix Group

Answer 1

A field is a matrix group if it satisfies:

1. If A, B ∈ G, then AB ∈ G
2. I_n ∈ G
3. If A ∈ G, then A is invertible and A^−1 ∈ G

Question 2

Matrix Group is a subgroup of…

Answer 2

General Linear Group, GL_n(F)

Question 3

The Orthogonal Group is

Answer 3

A Matrix Group!

Question 4

Permutation Matrix

Answer 4

Every Perm Matrix is Orthogonal

Question 5

Group acting on set X (Group Action)

Answer 5

A group acts on set X is there exists a group homomorphism phi: G -> Sym(X)

Question 6

G-Set

Answer 6

If we say X is a G-Set then that means the group G acts on set X

Question 7

Stabiliser of G_x

Answer 7

Let G be a group and X be a G-set, the stabiliser for G_x is a subgroup of G

Question 8

Orbit

Answer 8

When a group G acts on a set X (this process is called a group action), it permutes the elements of X. Any particular element X moves around in a fixed path which is called its orbit

Question 9

Congruence (Orbits)

Answer 9

Let X be a G-set, and let x, y ∈ X. Then we say that x is congruent to y modulo G if x ∈ G · y

Question 10

Congruence Modulo G

Answer 10

This is an Equivalence relation on X

Question 11

Equivalence Classes for Congruence Modulo G

Answer 11

They are the G-Orbits

Question 12

Transitivity (Orbits)

Answer 12

Let X be a G-set. We say the action of G on X is transitive if there is only one orbit for the action