Definitions

Question 1

Group Action

Answer 1

An action of a group G on a set X is a homomorphism ρ: G —> S(x)

Question 2

Orbit of x

Answer 2

If G acts on X, the orbit of a element x ∈ X is the subset G•x = {ρ(g)x | g ∈ G}

Question 3

Stabilizer of x

Answer 3

Suppose G acts on X and let x ∈ X, the stabilizer of x is G_x = {g ∈ G| ρ(g)x = x}

Question 4

G_x,y

Answer 4

G_x,y = {g ∈ G| ρ(g)x = y}

Question 5

Left Coset

Answer 5

Let g ∈ G and H ≤ G then gH = {gh | h ∈ H} is a left coset of H

Question 6

Right Coset

Answer 6

Let g ∈ G and H ≤ G then Hg = {hg | h ∈ H} is a right coset of H

Question 7

Conjugate

Answer 7

2 subgroup H, K of G are conjugate if ∃g ∈ G such that H = gKg^-1

Question 8

Transitive

Answer 8

Let ρ: G –> S(x) be an action of G on X. The action is transitive if ∀x, y ∈ X, ∃g ∈ G such that ρ(g)x = y

Question 9

Free

Answer 9

Let ρ: G –> S(x) be an action of G on X. The action is free if ∀x ∈ X, G_x = 1

Question 10

Effective

Answer 10

Let ρ: G –> S(x) be an action of G on X. The action is effective if ∀g ∈ G, ∃x ∈ X such that ρ(g)x != x

Question 11

Equivariant

Answer 11

Suppose G acts on 2 sets, X, Y. A map ∅: X –> Y is equivariant if ρ_Y(g)∅(x) = ∅(ρ_X(g)x), ∀g ∈ G, ∀x ∈ X

Question 12

Isomorphic

Answer 12

Suppose G acts on 2 sets, X, Y. A map ∅: X –> Y is isomorphic if there exists an equivariant map ∅: X –> Y which is a bijection

Question 13

Orbit Type

Answer 13

G acts on X transitively. The orbit type of X is the conjugacy class of the stabilizer of any x ∈ X. If G_x = H, we write the orbit type as (H)

Question 14

Euclidean Transformation

Answer 14

A Euclidean Transformation is an isometry of R^n, f:R^n –> R^n such that |f(x)-f(y)| = |x-y|, ∀x,y ∈ R^n

Question 15

Orthogonal

Answer 15

A matrix A is orthogonal if A^TA = I

Question 16

Seitz Symbol

Answer 16

(A | v) where A is a matrix and v is a translation. (A | v)x = Ax + v

Question 17

Glide Reflection

Answer 17

A glide reflection in the plane consists of a reflection in a line l followed by a translation parallel to that line

Question 18

Lattice

Answer 18

A lattice in R^n is a subset of the form

L={m1a1+ … + mnan | mi ∈ Z} = Z{a1, …, an} where {a1, …, an} is a basis of R^n

Question 19

T_G

Answer 19

Let G ≤ E(2). Then T_G = G n R^2 = ker(π|_G) = {translations in G}. T_G is the translation subgroup of G

Question 20

J_G

Answer 20

J_G = π(G) < O(2).

J_G is called the point group of G

Question 21

Wallpaper Group

Answer 21

A wallpaper group is any subgroup of E(2) with the following properties:

1) Its translation subgroup T_W is a lattice
2) Its point group J_W is finite

Question 22

Linear action of a group G on V

Answer 22

Let V be a finite dimensional vector space. A linear action of a group G on V is a homomorphism
ρ: G –> GL(V)
where GL(V) = the general linear group on V, the group of all invertible linear transformations of V

Question 23

Invariant

Answer 23

Let f: V –> R, f is invariant if f(g•v) = f(v), ∀v ∈ V, ∀g ∈ G