Lecture 5 Group actions

Transcription

1 Lecture 5 Group actions

2 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 H = H. (b) Check that xy 1 H for all x, y H.

3 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 H = 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(g 1 g 2 ) = ϕ(g 1 )ϕ(g 2 ). Example: ϕ : (R, +) (R >0, ) by ϕ(x) = e x. Non-example: ϕ : (R >0, ) (R, +) by ϕ(x) = x (inclusion).

4 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)

5 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!)

6 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.

7 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}

8 Group actions Goal: Build automorphisms of sets (permutations) by using groups.

9 Group actions Goal: Build automorphisms of sets (permutations) by using groups. Fact: Aut(A) = S A for a set A.

10 Group actions Goal: Build automorphisms of sets (permutations) by using groups. Fact: Aut(A) = S A 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 S X permutes the objects of X. 3. The invertible matrices move vectors around in a vector space. 4. The symmetric group S n can also move vectors in R n around when you map its elements to permutation matrices.

11 Group actions Goal: Build automorphisms of sets (permutations) by using groups. Fact: Aut(A) = S A 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 S X permutes the objects of X. 3. The invertible matrices move vectors around in a vector space. 4. The symmetric group S n can also move vectors in R n 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 S A.

12 Group actions: new language, same idea. Definition A group action of a group G on a set A is a map from G A (g, a) A 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.

13 Group actions: new language, same idea. Definition A group action of a group G on a set A is a map from G A (g, a) A 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 S A g σ g defined by σ g (a) = g a. In other words, given a homomorphism, you get an action, and vice versa.

14 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 S X acts on X by permutation. 3. The invertible matrices act on vector spaces. 4. The symmetric group S n also acts on R n by permutation matrices.

15 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 S X acts on X by permutation. 3. The invertible matrices act on vector spaces. 4. The symmetric group S n also acts on R n by permutation matrices. Any action of a group on a vector space (which is the same thing as a homomorphism of G into GL n ) is called a representation of G. In particular, the map G S A 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, σ g is just one of those permutation matrices (exactly one 1 in each row and column).

16 Some vocabulary: The trivial action is g a = a, i.e σ g = 1 for all g G.

17 Some vocabulary: The trivial action is g a = a, i.e σ g = 1 for all g G. If the map g σ g is injective, we say the action is faithful.

18 Some vocabulary: The trivial action is g a = a, i.e σ g = 1 for all g G. If the map g σ g is injective, we say the action is faithful. The kernel of an action is the set {g G g a = a a A}.

19 Some vocabulary: The trivial action is g a = a, i.e σ g = 1 for all g G. If the map g σ g 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.

20 Some vocabulary: The trivial action is g a = a, i.e σ g = 1 for all g G. If the map g σ g 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 R n by scaling: x (v 1, v 2,..., v n ) = (xv 1, xv 2,..., xv n ).

21 Some vocabulary: The trivial action is g a = a, i.e σ g = 1 for all g G. If the map g σ g 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 R n by scaling: x (v 1, v 2,..., v n ) = (xv 1, xv 2,..., xv n ). 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