Math 541

Unordered Concepts

If A is a set, the collection of all its subsets also forms a set, denoted as P(A) or 2A, called the power set.

The Cartesian product between A and B is the set consisting of all ordered pairs of elements in A and elements in B. We can write this as

A×B={(a,b):aA,bB}

Functions

A function or a map f between A and B is a relation between A and B, such that for any aA, there is a unique bB, such that (a,b)f . We denote this b as f(a). We often write such a function as

f:AB,af(a)

A is the domain, B is the codomain.

Relation of Functions

Injection X 1 2 3 Y D B C A Injection Surjection X 1 2 3 4 Y D B C Surjection BijectionX1234YDBCA

identity function

The identity function from A to itself is defined as idA:AA,aa. AKA, idA={(a,a):aA}A×A

If C=A, gf=idA, fg=idB, we call g the inverse of f,denoted as g=f1, and f to be invertible.

Group

Definition

A group is a pair (G,), where G is a set,:G×GG a map, such that:

(a,b) can also be written as ab or ab or ab. The identity can also be denoted as 0 or 1. When (G,) is a group we can also say “G is a group under ”.

Note: To show a function is well-defined is also proving that the closure of the group. For every pair of elements a,b in the group, the result of the group operation (often denoted as or +) on a and b must also be in the group. i.e. G×GG on (G,+). For any g,fG we have (f+g)(a)=f(a)+g(a) since f(a),g(a)G, f+gG. So, + is well-defined.

Note. Well-Defined means

  1. for some x,yG such that x=y implies f(x)=f(y).
  2. one-to-one of f(x)=f(y) implies x=y.

Abelian Group

If is commutative, i.e. for any a,bG, ab=ba, then we call it a commutative group or abelian group.

({e},(e,e)e). This is called the trivial group, often denoted as 0 or 1.

Example

For QQ there is a map action (composition action) f:qaq. Show Q is abelian group.

\begin{proof}
Suppose f(q)=aq, g(q)=bq are two elements in group, then for any qQ,

(fg)(q)=a(b(q))=abq=b(a(q))=(gf)(q)

Hence fg=gf is abelian.
\end{proof}

Permutation Group

A is any non-empty set, ({bijections from A to A }, ). This is called the permutation group or symmetric group. When A is a finite set of n elements this is denoted as Sn. Sn are all finite, and is abelian iff n2.

Example

1 2 3 4 1 2 3 4 1 2 3 4
(12)(14)(23)=(1423)

Homomorphisms

Definition

If (G,G) and (H,H) are two groups, f:GH satisfies that for any a,bG, f(aGb)=f(a)Hf(b), then we call f a homomorphism. If f is bijective then we call f an isomorphism.

Note: isomorphism: 1. Well-defined, 2. Bijective, Homomorphisms

Note: The two groups (G,) and (H,+) are isomorphic if there exists an isomorphism from one to the other written as

(G,)(H,+)

Sometimes write G=H

The set of isomorphisms from G to G (called “automorphisms”) is denoted as Aut(G).

For a group (G,G), a mapping f:GG is called automorphism if

  1. f is one-one.
  2. f homomorphic i.e. f(aGb)=f(a)Gf(b) for a,bG

Note: If H=G and =+ then the bijection is an automorphisms.

Property

If f is a homomorphism then:

  1. f sends the identity of G to the identity of H
  2. For any aG, (f(a))1=f(a1)

Isomorphism Theorem of Groups

Let G be a group, NG, f:GH a group homomorphism.
p:GQ a surjective homomorphism such that N=ker(p). Then:

  1. If Nker(f), then there is a unique homomorphism g:QH such that f=gp.
  2. If f is a surjection, so is g.
  3. If N=ker(f) then g is an injection.
  4. (Isomorphism Theorem of Group) f(G) is isomorphic to G/ker(f).

Subgroup

Definition

If HG contains eG and is also closed under product and inverse (for any a,bH ,abH ,a1H),then we call H a subgroup of G, denoted as HG.

Example

The group of integers Z under addition has subgroups of the form nZ for every integer n. For example:

Image and Kernel

If f is a homomorphism from G to H, then:

  1. The image (or range) of f, denoted as f(G), is a subgroup of H.
  2. f1({eH}), denoted as ker(f), called the kernel of f (and denoted as ker(f)), is a subgroup of G.
kerf={gG:f(g)=eH}forGH

Note: The kernel is a normal subgroup of G.

Example

Let G=(Z,+) and H=(Z5,+) which is modulo 5 under addition. The elements of H are {0,1,2,3,4}. Define a map f:GH by f(g)=gmod5.

The kerf is

kerf={gZ:gmod5=0}={,10,5,0,5,10,}

Interpretation:

Example 2

A group Z and a set {1,2,3} has one Z action:

u:Z×{1,2,3}{1,2,3}

defined as

u(n,x)=2+(1)n(x2)

Then the kernel of u is 2Z since only even number can make u(n,x) to be itself. i.e. u(2,x)=2+(1)2(x2)=x.

Normal Subgroup

A subgroup NG is called a normal subgroup, denoted as NG ,if for any aN , any gG , gag1N.

Group Action

Definition

Let G be a group, X a non-empty set. A left G action is a map f:G×XX, such that:

Reminder

In f(g,f(h,x))=f(gh,x), where gh is gh. For example,
For f:(g,x)xg , the relation should be

u(g,u(h,x))=u(gh,x)

where gh is represent as group action not simply gh.

Example

Let G=Z and X={1,2,3} so, u:Z×{1,2,3}{1,2,3} defined as map u(n,x)=n+x.
So, for any g,hG, any xX, we have

u(g,u(h,x))=u(g,h+x)=g+(h+x)=(g+h)+x=u(gh,x)

Note: we often write the result of a left G action as (g,x)gx or gx.

Let G be a group, then:

A map f:X×GX is called a right G action iff

Note

The automorphism of a G-set X is a bijective function f:XX such that for all fG and xX:f(gx)=gf(x). Which is also the G-equivariant (Math 541#^6f534d)

Permutation Representation

Let