Group Theory

An abelian group is a group in which the operation is commutative.

That is, for all elements (a) and (b),

[ a b = b a. ]

The name comes from the Norwegian mathematician Niels Henrik Abel.

Definition

A set (G) with an operation (\cdot) is an abelian group if:

Closure: [ a,b\in G \implies a\cdot b\in G ]
Associativity: [ (a\cdot b)\cdot c = a\cdot(b\cdot c) ]
Identity element (e): [ a\cdot e = e\cdot a = a ]
Inverse element: [ \forall a\in G,\ \exists a^{-1}\in G,\quad a\cdot a^{-1}=a^{-1}\cdot a=e ]
Commutativity: [ a\cdot b=b\cdot a. ]

Examples

Integers under addition

[ (\mathbb Z,+) ]

Identity: (0)
Inverse of (n): (-n)
Commutative because

[ m+n=n+m. ]

Real numbers under addition

[ (\mathbb R,+) ]

is abelian.

Nonzero real numbers under multiplication

[ (\mathbb R^\times,\cdot) ]

is abelian.

Circle group

[ S^1={z\in\mathbb C:|z|=1} ]

under complex multiplication is abelian.

Non-example

The group of invertible matrices

[ GL(n,\mathbb R) ]

for (n\ge 2) is generally not abelian because matrix multiplication does not commute:

[ AB \ne BA. ]

For example,

[ A= \begin{pmatrix} 1&1
0&1 \end{pmatrix}, \quad B= \begin{pmatrix} 1&0
1&1 \end{pmatrix} ]

satisfy (AB\neq BA).

Why abelian groups matter

Abelian groups are the simplest and most completely understood groups. They appear everywhere:

vector spaces (under addition),
Fourier analysis,
algebraic topology (homology groups),
number theory,
quantum mechanics.

In fact, every finite abelian group can be decomposed into cyclic pieces. This is the Fundamental Theorem of Finite Abelian Groups:

[ G \cong \mathbb Z_{n_1}\times \mathbb Z_{n_2}\times\cdots\times \mathbb Z_{n_k}. ]

For example,

[ \mathbb Z_{12} \cong \mathbb Z_3 \times \mathbb Z_4. ]

A useful way to think about it:

Abelian group = "addition-like" behavior ((a+b=b+a)).
Non-abelian group = "rotation-like" behavior, where the order of operations matters. For example, rotations of a cube in 3D form a non-abelian group.

A compact group is a group that is also a compact topological space.

More precisely:

A group is a set with an operation satisfying closure, associativity, identity, and inverses.
A topological group is a group equipped with a topology such that:
- multiplication ((x,y)\mapsto xy) is continuous, and
- inversion (x\mapsto x^{-1}) is continuous.
A compact group is a topological group whose underlying topological space is compact.

In simple terms, compactness means the space is “finite in extent” in a topological sense.

Examples

Finite groups
- Any finite group with the discrete topology is compact.
The circle group [ S^1={z\in \mathbb{C}: |z|=1} ] under complex multiplication.

S^1={z\in\mathbb{C}:|z|=1}
The group of orthogonal matrices [ O(n)={A\in M_n(\mathbb{R}) : A^TA=I} ] which is compact inside (\mathbb{R}^{n^2}).

O(n)={A\in M_n(\mathbb{R}):A^TA=I}
The unitary group [ U(n)={A\in M_n(\mathbb{C}) : A^*A=I} ]

Non-examples

((\mathbb{R},+)) is not compact.
((\mathbb{Z},+)) with the discrete topology is not compact.

Why compact groups are important

Compact groups play a central role in:

harmonic analysis,
representation theory,
quantum physics,
Lie theory.

A key property is the existence of a unique translation-invariant probability measure called the Haar measure.

If you'd like, I can also explain:

compact Lie groups,
why (O(n)) is compact,
Haar measure,
or the difference between compact and locally compact groups.

Теория групп. (Алексей Савватеев)

https://teach-in.ru/course/group-theory-manukhov

https://www.youtube.com/playlist?list=PLsdv2rpSb41YX3GXvKg7YQOwl-MVNkz9H

https://www.youtube.com/watch?v=ihoATq9jSlQ Лекция 1

https://www.youtube.com/watch?v=cEC7gPN441w Лекция 2

https://www.youtube.com/watch?v=m0Fz88VtTKc 3

https://www.youtube.com/watch?v=OgQSmqZABVY 4

https://www.youtube.com/watch?v=OraV8VbFzrQ 5

https://www.youtube.com/watch?v=obXICujbJMw 6

https://www.youtube.com/watch?v=ytRgMCC2DN0 7

Основы высшей алгебры и теории кодирования (1 курс, весна 2022) - Лектор: Вялый М.Н.

https://www.youtube.com/watch?v=aF0amljoZk4&list=PL4_hYwCyhAvZQoUsDe17ZGXIxcG3O84RG

https://publications.hse.ru/mirror/pubs/share/direct/433420486.pdf

Lie groups in physics

https://arxiv.org/pdf/2012.00834.pdf

https://www.youtube.com/watch?v=w-HygD3yLho

Lie Group

A Lie group (pronounced "Lee group", after Sophus Lie) is a mathematical object that is simultaneously:

A group (it has a multiplication, identity, inverses), and
A smooth manifold (locally it looks like (\mathbb{R}^n), and you can do calculus on it).

Moreover, the group operations must be smooth:

[ (x,y)\mapsto xy ]

and

[ x\mapsto x^{-1} ]

are differentiable maps.

Intuition

A Lie group describes continuous symmetries.

Examples:

Rotations of a circle
Rotations of 3D space
Translations of space
Lorentz transformations in relativity
Gauge symmetries in particle physics

Whenever a symmetry depends on continuously varying parameters, a Lie group is usually involved.

Example 1: Circle group (U(1))

The set

[ U(1)={e^{i\theta}: \theta\in\mathbb{R}} ]

forms a group under multiplication.

Geometrically it is a circle:

[ S^1. ]

Each element is determined by one parameter (\theta).

So:

Group structure: multiplication of complex numbers.
Manifold structure: a 1-dimensional circle.

Therefore (U(1)) is a 1-dimensional Lie group.

Example 2: Rotations in the plane

Every rotation can be written as

[ R(\theta)= \begin{pmatrix} \cos\theta & -\sin\theta
\sin\theta & \cos\theta \end{pmatrix}. ]

The set of all such matrices is

[ SO(2). ]

This is also a 1-dimensional Lie group.

Example 3: Rotations in 3D

The group

[ SO(3) ]

consists of all (3\times3) rotation matrices.

Each rotation depends on three independent parameters.

Thus:

[ \dim SO(3)=3. ]

This group describes the rotational symmetry of ordinary space.

Matrix Lie Groups

Many important Lie groups are groups of matrices:

Group	Meaning	Dimension
(GL(n,\mathbb R))	invertible real matrices	(n^2)
(SL(n,\mathbb R))	determinant (1) matrices	(n^2-1)
(O(n))	orthogonal matrices	(n(n-1)/2)
(SO(n))	rotations	(n(n-1)/2)
(U(n))	unitary matrices	(n^2)
(SU(n))	special unitary matrices	(n^2-1)

Lie Algebra

One of Lie's great discoveries was that a Lie group can be studied by looking at its behavior near the identity element.

The infinitesimal version of a Lie group is its Lie algebra.

For example:

[ SO(3) ]

has Lie algebra

[ \mathfrak{so}(3), ]

which describes infinitesimal rotations.

Physicists often work with generators rather than full group elements.

For rotations:

[ R(\theta)

e^{\theta J} ]

where (J) is a generator.

This is analogous to

[ e^{itH} ]

in quantum mechanics.

Why Lie groups matter in physics

They describe nearly every fundamental symmetry:

Physics	Lie Group
Electromagnetism	(U(1))
Weak interaction	(SU(2))
Strong interaction	(SU(3))
Rotations	(SO(3))
Relativity	Lorentz group
Quantum states	Unitary groups

The Standard Model gauge symmetry is

[ SU(3)\times SU(2)\times U(1). ]

Compact vs Non-compact Lie Groups

Examples of compact Lie groups:

(U(1))
(SO(3))
(SU(2))

Examples of non-compact Lie groups:

((\mathbb R,+))
(GL(n,\mathbb R))
Lorentz group

Compact Lie groups have finite volume (with Haar measure) and especially nice representation theory.

One-sentence summary

A Lie group is a group of continuous symmetries that is also a smooth geometric space, allowing algebra and calculus to be used together. It is the mathematical language of symmetry in geometry, physics, and differential equations.

The phrase "free parameters" is not a standard concept in abstract group theory, but it appears in several related contexts.

1. Free parameters in a Lie group

For continuous groups (Lie groups), the elements can be described by a certain number of continuous parameters.

Examples:

The circle group (U(1)): [ e^{i\theta} ] has one free parameter (\theta).
The rotation group (SO(3)): every rotation can be described by an axis (2 parameters) and an angle (1 parameter), so it has 3 free parameters.
The unitary group (U(n)) has (n^2) real parameters.

In this context, "free parameters" means the number of independent continuous coordinates needed to specify a group element.

2. Free generators in a free group

Perhaps you mean a free group.

A free group on generators

[ F(a,b,c) ]

has no relations except those required by the group axioms.

The generators (a,b,c) are sometimes informally called "free parameters" because they can be combined arbitrarily.

Example:

[ a^2 b^{-1} c a ]

is a valid element of (F(a,b,c)).

The only simplifications are things like

[ a a^{-1} = e. ]

3. Parameters in a group presentation

A group may be defined by generators and relations:

[ G = \langle a,b \mid a^2=e,\ b^3=e \rangle. ]

Here (a) and (b) are generators, and the relations constrain them.

If there are fewer relations, there is more "freedom."

4. Degrees of freedom in matrix groups

Consider

[ SO(2)= \left{ \begin{pmatrix} \cos\theta & -\sin\theta
\sin\theta & \cos\theta \end{pmatrix} \right}. ]

A (2\times2) matrix has 4 entries, but the orthogonality conditions force all entries to depend on a single parameter (\theta).

So (SO(2)) has one free parameter.

Similarly:

Group	Number of free parameters
(SO(2))	1
(SO(3))	3
(SU(2))	3
(U(n))	(n^2)
(SU(n))	(n^2-1)

If your question comes from physics (Lie groups, gauge theory, quantum mechanics), then "free parameters" usually means the dimension of the Lie group—the number of independent continuous variables needed to specify an element. If it comes from algebra, you may be referring to free groups and free generators.

LightCone

https://www.youtube.com/watch?v=wA8Wa2zAJ3E&list=PLnbH8YQPwKblIpRi0ARO2VadnMwntvF51&pp=iAQB

https://www.youtube.com/watch?v=wTdO_N0qGg4

https://www.youtube.com/watch?v=ajkof5THayg Лекция №1 по Теории групп. Смежные классы. Теорема Лагранжа. Богданов И.И.

https://www.youtube.com/watch?v=ajkof5THayg

https://www.youtube.com/playlist?list=PLi01XoE8jYoi3SgnnGorR_XOW3IcK-TP6