Group theory and related

An attempt ot get familiar with group theory from the internet.

Has got applications in Physics, Chemistry and beyond.

Found on a webpage related to chemistry:

Group theory is the mathematical application of symmetry to an object to obtain knowledge of its physical properties.

Rubik's cube can be represented using group theory concepts.

Field

A set for which the following relations are defined:

Most well-known example: ℝ (set of real numbers)

Many cryptographic protocols rely on fields.

Rings

Has two relations defined:

that satisfies the following axioms (ring axioms):

Groups

Image

A set associated with a binary operation that combines any two elements of the set to produce a third element in such a way that some conditions (called group axioms are satisfied).

(A, op)

Group axioms

Associativity

Changing order of evaluation of two adjacent operations of the same operator won't change the result.

x op (y op z) == (x op y) op z ∀ x, y, z ∈ A and op: A → A

Identity

***# Identity element For a binary relation R defined on a set A, the identity element i is an element such that

R(i, k) = k, ∀k ∈ A (left identity)

or

R(k, i) = k, ∀k ∈ A (right identity)

Two sided identity if i is both left identity and right identity.

Identity when the binary relation is addition is called additive identity.

Sometimes denoted as 0.

This could be something other than zero for stuff like groups and rings.

ie, when k + 0 = k, ∀k ∈ A

Invertibility

***# Inverse element For a binary relation R defined on a set A,

R(k, i) = e, ∃i ∈ A, ∀k ∈ A

where e is the identity element.

Example

Set of integers associated with the addition operation.

ie,

(ℤ, +)

Associativity

x + (y + z) == (x + y) + z ∀ x, y, z ∈ ℤ and +: ℤ → ℤ

Identity

The number 0 itself can be the identity element.

k + 0 = k, ∀k ∈ ℤ

Inverse

∀k ∈ ℤ, -k is the inverse element.

Abelian group

A group where the relation is commutative (in addition to being associative).

Eg: G(ℤ, +, 0) # ie, the set of integer with the addition operation as the relation and 0 as the identity element.

Finite groups

Groups whose underlying set is finite.

So groups on ℕ, ℤ, etc are not finite groups.

Symmetric groups

Example:

S₃:

The elements are:

ε = ⎡ 1  2  3 ⎤    α = ⎡ 1  2  3 ⎤    α² = ⎡ 1  2  3 ⎤
    ⎣ 1  2  3 ⎦        ⎣ 2  3  1 ⎦         ⎣ 3  1  2 ⎦

        ()               (1,3,2)              (1,2,3)



β = ⎡ 1  2  3 ⎤   αβ = ⎡ 1  2  3 ⎤   α²β = ⎡ 1  2  3 ⎤
    ⎣ 1  3  2 ⎦        ⎣ 2  1  3 ⎦         ⎣ 3  2  1 ⎦

       (2,3)             (1,3)                (1,2)

where the map is like the positions in first row being mapped to those of the second row.

(Here, αβ means α is applied first, and then β.)

Note: S₃ is a non-Abelian group. Eg: βα ≠ αβ (and βα is same as α²β).

sage: s3 = SymmetricGroup(3)
sage: s3
Symmetric group of order 3! as a permutation group

sage: list(s3)
[(), (1,3,2), (1,2,3), (2,3), (1,3), (1,2)]

The list(s3) output from sagemath is in cycle decomposition notation (aka disjoint cycle notation).

—

Sₙ (Symmetric group of degree n):

ε = ⎡   1     2  ..   n ⎤
    ⎣ α(1)  α(2) .. α(n) ⎦

where α(n) is the position to which position n is mapped.

Order(Sₙ) = n!

Permutation group

Cayley's theorem

Every group is isomorphic to some permutation group.

Legrange's theorem (group theory)

Given:

Then:

ℤ₆ = {0, 1, 2, 3, 4, 5}   
H  = {0, 2, 4}   
ℤ₃ = {0, 1, 2}   

ℤ₆/H = {0, 2, 4}

Monoid

A set associated with a binary relation where:

Effectively, monoids are semigroups with identity.

Or to put differently, groups where inverse elements aren't required.

Different kinds of monoids include:

Cyclic group

Group is cyclic if there's an element a ∈ G such G = {aⁿ | n ∈ ℤ}.

where a is said to be the generator.

Notation: <a> where a∈G is the cyclic subgroup of G generated by a.

Eg:

For (ℤ, +, 0), both 1 and -1 are generators.

ie, ℤ = <1> and ℤ = <-1>

Other examples:

- ℤ₈ = <1>
     = <3> = {3, 3*2, 3*3, 3*4, 3*5, 3*6, 3*7, 3*8}
           = {3,6,1,4,7,2,5,0}
     = <5> = {5, 5*2, 5*3, 5*4, 5*5, 5*6, 5*7, 5*8}
           = {5,2,7,4,1,6,3,0}
- ℤ₆ = <1> = <5>

—

Cyclic goupr => group can be generated by a single element.

Eg: 1 is generator of ℤ₆. ∵ we can get all elements of ℤ₆ by repeatedly doing the group operation on 1.

Fun facts:

Degree of a permutation group

Not the same as order of a group!

sage: s3 = SymmetricGroup(3)
sage: s3.degree()
3

sage: list(s3)
[(), (1,3,2), (1,2,3), (2,3), (1,3), (1,2)]

sage: s3.order()
6

Tips

A diagram

Monoid
  |
  | + inverse
  |
Group
  |
  | + add operation
  |
Ring
  |
  |
  |
Field

Closure

Consider a relation R on a set A where R may or may not have a property P (like reflexivity, symmetry, etc). If there is a relation S with the property P such that R ⊆ S, then S is the closure

For example, transitive closure of a relation R is the smallest transitive relation that contains R.

Reference: Discrete Mathematics by Rosen

Groupoid

(Related to category theory as well.)

A group with a partial function instead of the binary operation.

Setoid

A set, together with an equivalence relation.

(nlab link).

Comparison of group-like structures

From Wikipedia:

Totality Associativity Identity Division Commutativity
Semigroupoid - ✓ - - -
Small category - ✓ ✓ - -
Groupoid - ✓ ✓ ✓ -
Magma ✓ - - - -
Quasigroup ✓ - - ✓ -
Unital magma ✓ - ✓ - -
Semigroup ✓ ✓ - - -
Loop ✓ - ✓ ✓ -
Group ✓ ✓ - ✓ -
Monoid ✓ ✓ ✓ - -
Group ✓ ✓ ✓ ✓ -
Commutative monoid ✓ ✓ ✓ - ✓
Abelian group ✓ ✓ ✓ ✓ ✓

Cayley diagrams

aka Cayley graphs.

Graphical representation of a permutation group.

Kernel of group homomorphism ˡ

Given:

then

Info:

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