Download Lecture 5 Group actions

Lecture 5
Group actions
From last time:
1. A subset H of a group G which is itself a group under the
same operation is a subgroup of G.
Two ways of identifying if H is a subgroup or not:
(a) Check that hHi = H.
(b) Check that xy −1 ∈ H for all x, y ∈ H.
From last time:
1. A subset H of a group G which is itself a group under the
same operation is a subgroup of G.
Two ways of identifying if H is a subgroup or not:
(a) Check that hHi = H.
(b) Check that xy −1 ∈ H for all x, y ∈ H.
2. A homomorphism is a map of groups ϕ : G → H satisfying
ϕ(g1 g2 ) = ϕ(g1 )ϕ(g2 ).
Example: ϕ : (R, +) → (R>0 , ×) by ϕ(x) = ex .
Non-example: ϕ : (R>0 , ×) → (R, +) by ϕ(x) = x (inclusion).
Mathematical aside: morphisms in general
(Not in D & F, but for our own betterment)
In general in math, a (homo)morphism is just a map from one
mathematical object to another of its own kind, which obeys the right
rules (“structure-preserving”).
(Look up “category theory” if you ever want to feel like you’re doing all math, ever, all at once)
Mathematical aside: morphisms in general
(Not in D & F, but for our own betterment)
In general in math, a (homo)morphism is just a map from one
mathematical object to another of its own kind, which obeys the right
rules (“structure-preserving”).
(Look up “category theory” if you ever want to feel like you’re doing all math, ever, all at once)
Examples:
1. We just defined homomorphisms of groups
2. Linear transformations are morphisms of vector spaces
3. Any map from one set to another is a morphism of sets (no rules!)
Mathematical aside: morphisms in general
(Not in D & F, but for our own betterment)
In general in math, a (homo)morphism is just a map from one
mathematical object to another of its own kind, which obeys the right
rules (“structure-preserving”).
(Look up “category theory” if you ever want to feel like you’re doing all math, ever, all at once)
Examples:
1. We just defined homomorphisms of groups
2. Linear transformations are morphisms of vector spaces
3. Any map from one set to another is a morphism of sets (no rules!)
Endomorphisms are morphisms from something to itself.
Isomorphisms are bijective morphisms.
Automorphisms are bijective endomorphisms.
Mathematical aside: morphisms in general
(Not in D & F, but for our own betterment)
In general in math, a (homo)morphism is just a map from one
mathematical object to another of its own kind, which obeys the right
rules (“structure-preserving”).
(Look up “category theory” if you ever want to feel like you’re doing all math, ever, all at once)
Examples:
1. We just defined homomorphisms of groups
2. Linear transformations are morphisms of vector spaces
3. Any map from one set to another is a morphism of sets (no rules!)
Hom(X, Y ) = {ϕ : X → Y | ϕ is structure-preserving}
Endomorphisms are morphisms from something to itself.
End(X) = Hom(X, X) = {ϕ : X → X}
Isomorphisms are bijective morphisms.
Automorphisms are bijective endomorphisms. Aut(X) = {ϕ : X ↔ X}
Group actions
Goal: Build automorphisms of sets (permutations) by using groups.
Group actions
Goal: Build automorphisms of sets (permutations) by using groups.
Fact: Aut(A) = SA for a set A.
Group actions
Goal: Build automorphisms of sets (permutations) by using groups.
Fact: Aut(A) = SA for a set A.
Examples we already know:
1. The dihedral group permutes the set of symmetric states a
regular n-gon can occupy.
2. The symmetric group SX permutes the objects of X.
3. The invertible matrices move vectors around in a vector space.
4. The symmetric group Sn can also move vectors in Rn around
when you map its elements to permutation matrices.
Group actions
Goal: Build automorphisms of sets (permutations) by using groups.
Fact: Aut(A) = SA for a set A.
Examples we already know:
1. The dihedral group permutes the set of symmetric states a
regular n-gon can occupy.
2. The symmetric group SX permutes the objects of X.
3. The invertible matrices move vectors around in a vector space.
4. The symmetric group Sn can also move vectors in Rn around
when you map its elements to permutation matrices.
Basically, if A is the set we’re permuting, then the goal is to find
homomorphisms from G into SA .
Group actions: new language, same idea.
Definition
A group action of a group G on a set A is a map from
G×A →A
(g, a) 7→ g · a
which satisfies
g · (h · a) = (gh) · a
and
1·a=a
for all g, h ∈ G, a ∈ A. We say G acts on A.
Group actions: new language, same idea.
Definition
A group action of a group G on a set A is a map from
G×A →A
(g, a) 7→ g · a
which satisfies
g · (h · a) = (gh) · a
and
1·a=a
for all g, h ∈ G, a ∈ A. We say G acts on A.
Theorem
A group action is equivalent to a homomorphism
G → SA
g 7→ σg
defined by σg (a) = g · a.
In other words, given a homomorphism, you get an action, and vice versa.
Back to the same examples from before:
1. The dihedral group acts on the set of symmetric states a
regular n-gon can occupy by rotations and flips.
2. The symmetric group SX acts on X by permutation.
3. The invertible matrices act on vector spaces.
4. The symmetric group Sn also acts on Rn by permutation
matrices.
Back to the same examples from before:
1. The dihedral group acts on the set of symmetric states a
regular n-gon can occupy by rotations and flips.
2. The symmetric group SX acts on X by permutation.
3. The invertible matrices act on vector spaces.
4. The symmetric group Sn also acts on Rn by permutation
matrices.
Any action of a group on a vector space (which is the same thing
as a homomorphism of G into GLn ) is called a representation of
G. In particular, the map G → SA is called the permutation
representation associated to the given action because you’re
pretending that A is a basis for a vector space. Thought of this
way, σg is just one of those permutation matrices (exactly one 1 in
each row and column).
Some vocabulary:
∗ The trivial action is g · a = a, i.e σg = 1 for all g ∈ G.
Some vocabulary:
∗ The trivial action is g · a = a, i.e σg = 1 for all g ∈ G.
∗ If the map g → σg is injective, we say the action is faithful.
Some vocabulary:
∗ The trivial action is g · a = a, i.e σg = 1 for all g ∈ G.
∗ If the map g → σg is injective, we say the action is faithful.
∗ The kernel of an action is the set {g ∈ G | g · a = a
∀a ∈ A}.
Some vocabulary:
∗ The trivial action is g · a = a, i.e σg = 1 for all g ∈ G.
∗ If the map g → σg is injective, we say the action is faithful.
∗ The kernel of an action is the set {g ∈ G | g · a = a
∀a ∈ A}.
∗ The way we’ve been writing the action is called a left action.
Sometimes it’s better to write
a·g
means g is acting from the right.
Some vocabulary:
∗ The trivial action is g · a = a, i.e σg = 1 for all g ∈ G.
∗ If the map g → σg is injective, we say the action is faithful.
∗ The kernel of an action is the set {g ∈ G | g · a = a
∀a ∈ A}.
∗ The way we’ve been writing the action is called a left action.
Sometimes it’s better to write
a·g
means g is acting from the right.
More examples:
1. R acts on Rn by scaling:
x · (v1 , v2 , . . . , vn ) = (xv1 , xv2 , . . . , xvn ).
Some vocabulary:
∗ The trivial action is g · a = a, i.e σg = 1 for all g ∈ G.
∗ If the map g → σg is injective, we say the action is faithful.
∗ The kernel of an action is the set {g ∈ G | g · a = a
∀a ∈ A}.
∗ The way we’ve been writing the action is called a left action.
Sometimes it’s better to write
a·g
means g is acting from the right.
More examples:
1. R acts on Rn by scaling:
x · (v1 , v2 , . . . , vn ) = (xv1 , xv2 , . . . , xvn ).
2. Any group G acts on itself (let A = G) in several ways:
left regular action: g · a = ga
right multiplication: g · a = ag −1
conjugation: g · a = gag −1