Group theory

Group theory is an area of ​​abstract algebra dedicated to the study of an algebraic structure called group structure. These mathematical objects are very useful in fields of science such as physics or chemistry. Many symmetries of special relativity or quantum mechanics can be defined through the algebraic group structure.

The objective of this publication is to give the reader an introduction to the groups and what applications they have in science.

Group theory was born to study the solutions of algebraic equations through radicals. Expressions were obtained for the resolution of polynomial equations of degree 2, degree 3 and degree 4. However, the main problem was knowing if for degrees greater than 4 there were radical solutions to the polynomial equations.

In the 19th century, a famous mathematician named Niels Henrik Abel demonstrated that if there were radical solutions to polynomial equations of degree 5 or higher, we would never find them.

Later, the young Evariste Galois generalized this proof using the famous Galois theorem.

This theorem achieved a union between group theory and body theory and could be used to prove solutions of polynomial equations.

Therefore, the first mathematician to define the concept of a group was the Frenchman Galois.

Later mathematicians such as the Norwegian Sophus Lie extended the notion of group for use in analytical problems.

Today group theory is implicit in many areas of mathematics, physics or chemistry.

A group is an algebraic structure made up of a set of elements and a binary operation. This operation has to satisfy 4 properties. If they are met, then the set of elements forms an algebraic structure. A group.

The operation has to be internalized. That is, if we operate on two elements of the group, the result has to be another element of the set.

The elements of the set must satisfy the associative property.

There must also be a neutral element. That is, if we operate any element with the neutral, the final result is the element.

Finally we need the existence of a symmetrical element. That is, if we take an element of the group and operate it with its symmetry, we obtain the neutral element.

Have you not understood anything? Let's look at an example!

For the example we will use one of the most common groups: the additive group of reals. In this group the set of elements are the real numbers and the operation is the sum.

The elements of this group fulfill the associative property since:

a+(b+c) = (a+b)+c where a,b and c are real numbers.

There is also a neutral element, 0. a+0 = a

Each element a has a symmetric element (-a) since if we operate them we obtain the neutral element, 0: a+(-a) = 0.

The addition in the additive group of reals is an internal operation since if we add any two real elements we obtain another real number.

Indeed, the sum operation meets all the requirements. Therefore, the set of reals together with the sum operation forms an algebraic group structure.

If the operation is also commutative, that is, a+b = b+a, then the group is abelian.

Finite groups are those in which the set that forms it has a finite number of elements. Some examples are symmetric groups or cyclic groups.

Infinite groups are described analogously to finite groups. However, in this case the set has an infinite number of elements. A typical infinite group is the general line group formed by the set of invertible matrices of degree n defined on a field K. GL(n,K).

Many of these groups can take on additional structure. If group operations are smooth, continuous and regular they can acquire structures of topological spaces.

The famous Lie groups are born here. These are groups with an additional manifold structure (locally flat topological space). The properties of these types of objects are very interesting, having many uses in quantum physics or relativistic physics.

In short, Lie groups are algebraic group structures that, due to their algebraic properties, acquire a topological space structure called variety.

Lie groups are very useful in physics since they describe sets of symmetries in nature.

And what does this mean?

Well, there is a very famous theorem called Noether's theorem formulated by the fabulous mathematician Emilie Noether.

This says that any conserved magnitude in physics has a symmetry associated with it. For example, energy is conserved thanks to temporal symmetry. Linear momentum is conserved due to a translational symmetry and angular momentum is conserved since there is a rotational symmetry.

Next we will go into more detail on the use of group theory in theoretical physics.

Quantum field theories are Gauge theories which describe fundamental forces and interactions through certain symmetries.

Have you already guessed it?

Indeed, these symmetries are described through Lie groups. Let's look at some examples:

The strong nuclear interaction is the force that holds protons and neutrons together in the atomic nucleus.

The energy of the field associated with the strong interaction is described by the Lagrangian of the field.

It turns out that there are some symmetries of the SU(3) group that leave the Lagrangian of this field invariant. That is, if we apply elements of SU(3) to the energy, it remains invariant.

SU(3) is the Lie group formed by the unitary matrices of dimension 3 with determinant 1. By Noether's theorem, this Lagrangian has an associated conserved magnitude. In the case of the strong interaction, this magnitude is called color charge and is responsible for keeping the nuclei well united.

Fascinating right?

Another example of a quantum field with associated symmetries is the field of quantum electrodynamics, responsible for electromagnetic interaction.

This gauge theory has a U(1) symmetry associated with it.

The main role of symmetry groups in quantum mechanics is the description of spin. We have often heard that spin is a quantum property with no classical analogue that can be imagined as a rotation without the particle actually rotating (as we classically understand the rotation of a body).

In quantum mechanics we define spin as operators that act on wave functions in a Hilbert space. These operators transform like rotations. For this reason we can relate the property of spin as a rotation.

These rotations belong to the special unitary symmetry group SU(2), which is still a Lie group.

Special relativity tells us, in a simple way, that two observers can observe different realities of the same phenomenon. For example, if we travel on a train that is going at a speed of 150 km/h and we throw a ball at 10 km/h, for us the ball has a speed of 10 km/h.

However, for an observer outside the train the speed of the ball is 10 km/h plus the speed of the train, a total of 160 km/h.

As we see, there are different realities depending on the conditions of the observer. Therefore, we can conclude that reality is relative.

In physics, these “different realities” are mathematically related through Lorentz transformations.

Therefore, what one observer sees is equal to what the other observer sees multiplied by a Lorentz transformation.

It turns out that all Lorentz transformations form a group under the matrix multiplication operation.

Group theory is very useful in the field of theoretical chemistry. Molecules have certain symmetries that make it easier to calculate certain properties in addition to intuiting important characteristics.

When we do a quantum-level study of chemical molecules, we cannot exactly solve the Schrödinger equation since it is a polyelectric system.

To approximate the solution we need to solve complex double integrals that have a very high computational expense. In many cases, we can use symmetry to simplify these integrals and know if their result is zero.

In this way, thanks to the symmetries that are explained through group theory we can save a lot of calculation time.

Group theory, as it could not be otherwise, also has many applications in mathematics itself. As an example, I want to give you the use that group algebra has within topology.

Topology is a branch of mathematics that studies the shape of objects and surfaces. At a topological level, a donut is equivalent to a coffee cup, since we can deform one object into another without cutting or gluing, just deforming.

This transformation of the object is called the mathematical name of homeomorphism. Homeomorphisms can be very complex to study. This is where the role of abstract group algebra comes in. We can use groups to study these homeomorphisms in a much simpler way.

This area of ​​mathematics that uses group algebra to study topologies is called algebraic topology and has many uses within mathematics and physics.

Surely at some point in your life you have tried to solve this famous cube. There are many different versions of the Rubik's cube. The typical one that everyone has is 3x3x3 but there are also 4x4x4, 5x5x5 and even 13x13x13.

Well, it turns out that this famous toy has hidden the ins and outs of group theory. The movements of this cube are permutations and the set of permutations form a group. Therefore, an in-depth study of this toy can be done through group algebra.