MENTAL vs. GROUP THEORY

"Group theory is the supreme art of mathematical abstraction" (James R. Newman).

"Group theory is the 'official' language of symmetries" (Mario Livio).

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Group Theory
Group theory was developed by Évariste Galois, at the beginning of the 19th century, as a consequence of the study of the resolution of the fifth degree equation [Livio, 2005].

The modern definition of group is due to Arthur Cayley, who in 1854 established the axioms of its definition. In 1878, Cayley discovered that every group (regardless of the type of its elements and the operation defined between them) is isomorphic to a group of permutations. The group concept then reached its highest level of abstraction and understanding.

Group theory has been applied to numerous branches of mathematics and physics, to the point of being considered a true unifying theory. Seemingly unrelated areas (theory of algebraic equations, geometries, number theory, etc.) are unified by the underlying group structure.


The definition of group

A group is a set G of elements having the following properties:

1. G inner operation.
   There is defined a binary (•) operation (or law of internal composition) that matches any ordered pair of elements (equal or different) of G another element of G. Actually, it is a function g of two arguments (elements of GG. Formally, g: G² → G.
   
   
2. Associativity.
   The inner operation is associative: x•(y•z) = (x•y)•z.
   
   
3. Neutral element.
   There exists a neutral element e from the left in G such that e•x = x for every element x of G.
   
   
4. Inverse element.
   For every element x of G, there exists a left inverse element x' such that x'•x = e. The notation x⁻¹ is also commonly used for the inverse element.

If in addition the inner operation has the commutative property (x•y = y•x), the group is called "abelian".

With these 4 axioms it is proved:

1. That the element neutral on the left is also neutral on the right. That is, for all x of G, it is satisfied that e•x = x•e = x, and that neutral element e is unique.
   
   
2. That the inverse element on the left is also the inverse element on the right, i.e., for all x of G, it is satisfied that x•x' = x'•x = e, and that the inverse element of every element x of G is unique.

There can be finite or infinite groups. The number of elements of a finite group G is called the "order of the group" and is denoted by |G|.

• An example of a finite group is Z_n = {0, 1, 2, .... , n} with the addition operation modulo n. The neutral element is 0. The inverse element of m is m' = n−m (every element and its inverse add up to n). If, for example, n=2, we have: 0'=2, 1'=1 and 2'=0. The group is abelian because the sum is commutative.
  
  
• An example of an infinite group is the set Z of integers with the inner operation of addition. The neutral element is 0 and the symmetric element of x is −x. The sum is associative and commutative. The group is abelian.
  
  
• Another example of an infinite group is the set R^* of the nonzero real numbers (R without zero) under the multiplication operation. The neutral element is 1 and the inverse of a number r is 1/r. The group is associative and commutative. The group is also abelian.

There is a single one•element group, I = {e}, which satisfies e•e = e. In this case, e is the neutral element and its own inverse. This group is called the identity.

There is only one group of 2 elements, reflected in this table:

•	e	a
e	e	a
a	a	e

Fulfilled: e² = a, a² = e, e•a = a•e = a, where we use exponential notation: x² = x•x.

There is also only one group of 3 elements:

•	e	a	b
e	e	a	b
a	a	b	e
b	b	e	a

And there are two groups of 4 elements.


Cyclic groups

A cyclic group is a group that is generated by a single element. That is, there is an element a of G (called the "generator" of G) such that all other elements of G (excluding the neutral element e) are expressed as powers of a: a² = a•a, a³ = a•a•a, etc. , the last power being equal to the neutral element. Cyclic groups can be finite or infinite.

An example of a finite cyclic group of 3 elements is defined by the following table:

•	e	a	a2
e	e	a	a2
a	a	a2	e
a2	e	e	a

That is, the unique 3•element group is cyclic and abelian. In general, if we represent aⁱ as a_i and e as a₀, we have that a_i •a_j = a_k, siendo < i>k = i+j (modulo n), where n = |G|.

The group Z_n is also an example of a cyclic group, where 1 is the generator of the group.


Symmetry groups

Symmetry groups are considered the "parents" of all groups, and are the ideal framework for studying the fundamental concept of symmetry.

A symmetry group S_n is the set of permutations (which are sequences) of n elements, and whose number is n! For example, for two elements, a₁ and a₂, the permutations are P₀ = ( a₁, a₂) and P₁ = (a₂, a₁). One goes from one to the other by reversing the order. The permutation P₀ is the neutral permutation, which leaves the permutation as it is. The table of products of both permutations is as follows:

•	P0	P1
P0	P0	P1
P1	P1	P0

La expresión P_i•P_j indicates to apply first the permutation P_j and then the P_i. P₀ and P₁ are inverse permutations of themselves. Therefore, their squares are the permutation identity:

P₀² = P₁² = P₀

Subgroups, normal subgroups and simple groups

A subgroup F of a group G (F &rise; G) is a subset of G that has group structure with respect to the same operation defined in G. According to Lagrange's theorem, if F is a subgroup of a finite group G, |F| is a divisor of |G|. Two trivial subgroups are the group G itself and the identity group I.

A normal subgroup N of a group G is a subgroup such that for every element x of G and for every element n of N it is satisfied that xn = nx. Another way of putting it is that the subgroup N is invariant by conjugation: xnx' ∈ N.

If a group G has as normal groups only the two trivial ones, the group G itself and the identity group I, then the group is said to be simple. That is, simple groups do not contain normal proper subgroups.

Any finite group can be constructed by a collection of finite simple groups; these are the "atomic" constituents of the groups.

All abelian groups, except those of prime order, have normal proper subgroups. All abelian groups of prime order are simple.

The qualifier "simple" to a group does not indicate that it is simple; it only indicates that it is not decomposable into subgroups. The simplest simple groups are abelian groups. The simple groups were discovered by Galois when investigating the possible solutions of the fifth degree equation.

Simple groups behave like prime numbers: they cannot be decomposed into normal proper subgroups. But, unlike prime numbers, their number is finite. The largest finite simple group, called "the monster", is of order.



Isomorphic groups

Two groups are isomorphic when a biunivocal correspondence f can be established between the elements of both groups, so that internal relations are preserved: if x•y = z, then f(x)•f(y) = f(z).

According to Cayley's theorem, every finite group is isomorphic to a symmetry group. That is, the structure of a group of order n is the same as a group of permutations of the same order n. Therefore, all groups are grounded on the symmetry groups.


Alternating groups

An alternating group A_n is a symmetry group constituted by the even permutations of n elements. Two elements are in inversion when their order differs from the main permutation. An even permutation is a permutation with an even number of inversions. An odd permutation is a permutation with an odd number of inversions. Even and odd permutations have n!/2 elements. For example, in the permutation 2134, elements 1 and 2 are inverted with respect to the main permutation 1234. In 2431 there are 4 inversions: 2•1, 4•3, 4•1 and 3•1.

An example of an alternating group (A₄) generated by 4 elements is:

1234	2143	3124	4132
1342	2314	3241	4213
1423	2431	3412	4321

An alternating group A_n is a simple group when n≥5. This is precisely the reason why the fifth degree equation (the quintic) is unsolvable by radicals. A₅ is the smallest of the simple alternating groups and its order is 5!/2 = 60.


Erlangen's program

The Erlangen program is a research program reflected in a paper published by Felix Klein in 1872 when he was a professor in Erlangen (Germany). In it he proposed a new approach to the study of geometry.

The emergence of new non•Euclidean geometries and the irruption of algebraic and analytical methods as opposed to synthetic and intuitive conceptions, made the foundations of traditional geometry questionable. Klein proposed to formally define each type of geometry by a set of transformations that leave objects invariant, without making any reference to these geometric objects. Each specific geometry is defined by certain properties that do not change (are invariant) when a type of transformations is applied to them. That set of transformations have group structure under the operation of composition of transformations.

For example, Euclidean geometry is the group of symmetries, rotations and translations (metric group). Affine geometry is the group of translations. Projective geometry is the group of projections. Topology is the group of continuous transformations.

Klein's article was a milestone in geometry and in mathematics in general. With Klein, the different geometries could be rigorously defined.


Polynomial Equations of One Variable
Vertical relationships between roots and coefficients

A polynomial equation of one variable has the form

a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₁x + a₀ = 0

where a_i are real or complex numbers and n is the degree of the equation. Since it can be divided by a_n, we can consider that a polynomial equation has the form

xⁿ + a_n−1xⁿ⁻¹ + ... + a₁x + a₀ = 0

According to the fundamental theorem of algebra (proved by Gauss), in a polynomial equation there are n solutions or roots r₁, ... , r_n. A polynomial equation is a shallow expression, so, in general, it is very difficult to solve it, that is, to obtain its roots. It is easier to start from its deep structure and, applying the principle of descending causality, to obtain the surface expression. The deep structure of the polynomial equation is

(x − r₁) ... (x − r_n) = 0

There may be repeated roots, as in the equation of degree 5 (x−a)²(x−b)³, where there are 2 roots equal to a and 3 roots equal to b.

Let's see how the surface equation develops as we increase the degree of the equation from the initial degree n=1. We are going to use the notation a_i,j to refer to the coefficient of x^j of the equation of degree i.

• degree 1.
  x + a_1,0 = 0
  (x − r₁) = 0
  a_1,0 = −r₁
  
  
• Grade 2.
  x² + a_2,1x + a_2,0 = 0
  
  (x + a_1,0)(x • r₂) = 0
  
  x² + (a_1,0 − r< sub>2)x − a_1,0r₂ = 0
  
  a_2,1 = a_1,0 − r< sub>2 = −r₁ − r₂
  
  a_2,0 = −a_1,0r₂ = r₁r₂
  
  
• Grade 3.
  x³ + a_3,2x² + a_3,1x + a_3,0 = 0
  
  (x² + a_2,1x + a_2,0)(x • r₃) = 0
  
  a_3,2 = a_2,1 − r₃ = − r₁ − r₂ − r₃
  
  a_3,1 = a_2,0 − a_2,1r₃ = r ₁r₂ + r₁r₃ + r₂r₃
  
  a_3,0 = −a_2,0r₃ = −r₁r₂r₃
  
  
• Grade 4.
  x⁴ + a_4,3x³ + a< sub>4,2x² + a_4,1x + a_4,0 = 0
  
  (x³ + a_3,2x² + a_3,1x + a_3,0)(x • r₄) = 0
  
  a_4,3 = a_3,2 − r₄ = − r< sub>1 − r₂ − r₃ − r₄
  
  a_4,2 = a_3,1 − a_3,2r ₄ = r₁r₂ + r₁r₃ + r₁r₄ + r₂r₃ + r₂r₄ + r₃r₄
  
  a_4,1 = a_3,0 − a_3,1r₄ = a_2,0 − a_2,1r₃ = −r< sub>1r₂r₃ − r₁r ₂r₄ − r₂r₃r₄
  
  a_4,0 = −a_3,0r₄ = r₁r₂r₃r₄

As can be seen, the structure of a polynomial equation is fractal because all equations have the same structure, i.e., the same mechanism is applied at each level starting from the immediately lower degree. Therefore, to obtain a solvable equation of degree n, we choose any values for r₁, ... , r_n. From them, the coefficients for the degrees 1, ... , n.

For example, for n=3 and r₁=1, r₂=3, r₃=5, we have:

a_1,0 = −r₁ = −1
a_2,1 = a_1,0 − r₂ = −1 • 3 = −4
a_2,0 = −a_1,0r₂ = 1−3 = 3
a_3,2 = a_2,1 − r₃ = −4 • 5 = −9
a_3,1 = a_2,0 − a_2,1r₃ = 3 + 4·5 = 23
a_3,0 = −a_2,0r₃ = − r₁r₂r₃ = −1·3·5 = −15

The equation is x³ − 9x² + 23x − 15 = 0 which equals (x • 1)(x • 1)(x • 3)(x • 5) = 0

Instead of performing a recursive process, the coefficients can also be calculated directly by the general formula

a_n,i = (−1)ⁿ⁺¹P_n,i

being P_n,i the sum of the products of n roots taken from i in i.

• Grade 1.
  x • r₁ = 0
  
  
• Grade 2.
  x² − (r₁ + r₂)x + r₁r₂ = 0
  
  
• Grade 3.
  x³ − (r₁ + r₂ + r₃)x² + (r₁r₂ + r₁r₃ + r₂r ₃)x − r₁r₂r₃ = 0
  
  
• Grade 4.
  x⁴ − (r₁ + r₂ + r₃ + r< sub>4)x³ + (r₁r₂ + r₁r ₃ + r₁r₄ + r₂r₃ + r₂r₄ + r₃r₄)x² − (r₁r₂r₃ + r₁r₂r₄ + r< sub>2r₃r₄)x + r ₁r₂r₃r₄ = 0

When all the roots are equal (r), the polynomial equation has the form

xⁿ − C_n,1rx^{n −1} + C_n,2r₂x² − . .. − C_n,n−1r_n−1 x + C_n,nr_n = 0

where C_n,i is the number of combinations of n elements taken from i in i. The equation is equivalent to (x−r)ⁿ = 0.

From the dimensional point of view, all the terms have dimension n, with 1 being the dimension of each of the roots. There is symmetry in the coefficients: entre a_n,j y a_{n,< i>n−j}, from j=1 to j=n−1. If n is even, there are n/2 pairs of dual coefficients. If n is odd there are (n−1)/2 pairs of dual coefficients and there is one coefficient (the central one) that is dual of itself.

Therefore, the coefficients of the equation are manifestations of the roots. The roots are encoded or encrypted in the coefficients of the equation. There is a connection between the deep (the roots) and the shallow (the coefficients). The roots can permute (there are n! permutations possible) but the corresponding coefficients are invariant.


Horizontal relations between roots

In the second degree equation x² + a_2,1x + a_2,0 = 0 its two roots are:

r₁ = (−a_2,1 + √(a_2,12 − 4a_2,0))/2
r₂ = (−a_2,1 − √(a_2,12 − 4a_2,0))/2

These two roots satisfy the horizontal relations

r₁ + r₂ = −a_2,1 r₁r₂ = a_2,0

In these relations, the roots are interchangeable. They can be permuted and the relationships remain valid.

In the equation x⁴ = 4, its roots are: r₁ = 2, r₂ = −2, r ₃ = 2i, r₄ = −2i. The horizontal relations are satisfied

r₁ + r₂ = 0   r₃ + r₄ = 0

In this case, the 4 roots are not interchangeable. Only r₁ and r₂ are interchangeable, as well as r₃ and r₄. Of the 4! = 24 possible permutations of the roots, there are only 4 that preserve these relations:

r₁ r₂ r₃ r₄
r₂ r₁ r₃ r₄
r₁ r₂ r₄ r₃
r₂ r₁ r₄ r₃

Galois group

The Galois group associated to a polynomial equation of degree n is the largest of the subgroups of the group of permutations of the n elements such that they make the horizontal relations between the roots invariant. In other words, the 4 basic arithmetic operations (addition, subtraction, multiplication and division) are preserved. The Galois group indicates the "symmetry profile" of a polynomial equation. If there are no relations between the roots, the Galois group is S_n, the set of all permutations of n elements.

Galois proved that, for any degree n, it is always possible to find equations for which the Galois group is all S_n. In other words, in any degree there are equations that possess the maximum possible symmetry.

Galois created the concept of group of symmetries to solve the problem of polynomial equations and to discover why the fifth degree equation (the quintic) could not be solved by radicals (square root, cube root, etc.). Solutions with radicals exist only if and only if the corresponding Galois group has a particularly simple structure. This is called a "solvable group". In the equations of 2nd, 3rd and 4th degree (quadratic, cubic and quartic) the Galois groups are always solvable, so that the solutions are expressible by radicals. But the quintic symmetry group is not solvable, so it cannot be expressed by radicals.

From the properties of a Galois group one can determine whether or not an equation can be solved by radicals (square root, cube root, etc.), without needing to know the roots of the equation.

Galois used the following definitions:

• Normal group N of a group G.
  Is a group such that if n is an element of N (n∈N), then xnx⁻¹ also belongs to N (xnx⁻¹∈N) for every member x of the group G. It is said that N is a conjugacy invariant subgroup.
  
  The groups I = {e} and G are always trivial normal subgroups of G, since.
  
  
  xex⁻¹ = xx⁻¹ = e ∈ I
  xnx⁻¹ ∈ G
  
  If I and G are the only normal subgroups of G, then G is simple.
  
  All subgroups S of an abelian group are normal groups, since xnx ⁻¹ = nxx⁻¹ = n ∈ S.
  
  
• Resolvable group.
  Given a finite group G, we choose the largest normal subgroup, N₁. This normal subgroup can in turn contain normal subgroups, the largest one is chosen again, N₂. And so on, creating a chain of hierarchical normal subgroups until reaching the identity subgroup I (formed only by the neutral element):
  
  
  G ⊃ N₁ ⊃ N₂ ⊃ ... ⊃ I
  
  The orders of the subgroups are always divisors of the immediately preceding hierarchical one. A group G is resolvable if all the composition factors formed by the maximal descending normal subgroups are prime numbers.
  
  
• Solvable equation.
  A polynomial equation is solvable (i.e., the roots are expressible as radicals) when the corresponding Galois group is solvable.
  
  In the quintic equation the order of the group of permutations of its roots is 5! = 120. On decomposing it into normal subgroups, a normal subgroup of order 60, which is not prime, appears. Therefore, the quintic is not solvable by radicals.
  
  Each normal subgroup establishes the horizontal relations between its roots.

Group Theory vs. MENTAL
The group as a mathematical archetype

It has been suggested that the mathematical concept of group (in particular, symmetry groups) can be considered a mathematical archetype of order or structure. It would play a role analogous to that of symmetry in physics.

What can be stated is that this concept is closely connected to the topic of archetypes of consciousness for several reasons:

• It is connected with the number 3, the number of consciousness, since the operation defined in a group is based on a function of two input arguments that produces a third element as a result.
  
  
• It harmonizes opposites or solves the problem of opposites by uniting opposites through the neutral element.
  
  
• The neutral element being the inverse of itself, the neutral element symbolizes non•duality, undifferentiated consciousness and also wholeness.
  
  
• It is a self•sustaining structure, that is, it sustains itself by the relationships between its elements. Two elements "produce" a third. In turn, each of the "producing" elements is produced by another pair of elements. Thus until all the elements of the whole are covered, producing a self•sustaining, ordered, coherent, self•sufficient and self•functional structure.
  
  
• The 4 axioms of the definition of a group have been compared with the 5 axioms of Euclid's geometry. The group axioms would be the axioms of consciousness. This conjecture is based on the fact that they are simple axioms, of maximum conceptual economy and of maximum philosophical generality.

Group theory vs. MENTAL

There are differences and analogies between group structure and MENTAL primitives:

• Level of abstraction.
  Group theory is not the supreme level of abstraction. The supreme abstraction is achieved with universal semantic primitives.
  
  
• Archetypes.
  Group structure is a mathematical archetype, but it is not an archetype of consciousness. The primitives of MENTAL are primal archetypes, archetypes of consciousness. Group theory is, therefore, a second•order archetype.
  
  
• Deep structure.
  The deeper a concept is, the greater are its manifestations. In the case of group structure, its manifestations are very numerous, indicating that it is indeed a deep structure. Its manifestations are the particular groups that appear in numerous domains. In mathematics it appears in the natural numbers under the operation of addition, the real numbers (without zero) under the operation of multiplication, permutations, geometric rotations, equation solving, braids, etc. In quantum physics, groups have been used to classify elementary particles and to predict the existence of quarks.
  
  MENTAL is the deepest structure and the source of all possible expressions.
  
  
• Duality.
  In group structure, duality manifests as pairs of inverse (or symmetrical) elements. In MENTAL, duality manifests itself in primitives and in language features as integral union of opposites or duals (descriptive and operative, quantitative and qualitative, analytic and synthetic, etc.).
  
  
• Type of structure.
  In group theory, structures are static. In MENTAL, structures can be dynamic.
  
  
• Derived structures.
  Other structures (ring, field, etc.) can be defined from the group structure. In MENTAL too, since derivative operations and all kinds of expressions in general can be defined.

Definition of group in MENTAL

The definition of group G can be done by symbolizing the inner operation in an infix form (between the two arguments) or as a prefix of the two arguments. Here we will use the prefix form, as a function of name g.

1. Internal operation g.
   ⟨( x∈G → y∈G → g(x y)∈G )⟩
   
   
2. Neutral element e.
   ⟨( x∈G → e∈G → (g(x e) = x) )⟩
⟨( g(x e) = g(e x)) )⟩
   
   
3. Inverse element x' of x.
   ⟨( x∈G → x'∈G → e∈G → (g(x x') = e) )⟩
⟨( g(x x') = g(x' x) )⟩
   
   
4. Associativity.
   ⟨( x∈G → y∈G → z∈G → (g(x y) z) ≡ g(x g(y z))) )⟩

For an abelian group, additionally,

⟨( x∈G → y∈G → (g(x y) ≡ g(y x)) )⟩

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Addenda
Elementary symmetric polynomials

The elementary symmetric polynomials are those that are formed by the sum of products of different variables and that do not vary by permuting their variables. For example, x+y and xy are elementary symmetric polynomials. Neither are 3x+y nor xy+z.

The fundamental theorem of symmetric polynomials (Lagrange's) states that any symmetric polynomial of n variables can be expressed as a polynomial of elementary symmetric polynomials. For example,

x² + y² = (x + y)² • 2xy = α² • 2β
x³ + y³ = (x + y)³ • 3xy(x + y) = α • 3αβ

where α = x + y   and   β = xy.

As we have seen, any polynomial of degree n² can be expressed by elementary symmetric polynomials P_n,i of the n roots of the polynomial equation.


Noether's theorem

In physics, symmetry was used by Emmy Noether to prove the theorem that bears his name and is a central result in theoretical physics. This theorem states that any differentiable symmetry coming from a physical system has a corresponding conservation law. To each continuous symmetry corresponds a conservation law and vice versa.


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• Fresan, Javier. Hasta que el álgebra os separe. La teoría de grupos y sus aplicaciones. RBA Coleccionables, El mundo es matemático, 2011.
  
  
• Livio, Mario. La ecuación jamás resuelta. Cómo dos genios matemáticos descubrieron el lenguaje de la simetría. Ariel, 2007.
  
  
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