# Binomial equations & groups

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## Roots of the binomial equation

### Binomial equation

The equation of the form xk − 1 = 0 is called binomial equation. Solution of binomial equation consists of divisors of the expression xk−1. The number k is called order of binomial equation. When the order kεP, we say that the binomial equation is simple order.

Exponents of individual members in the enrollment of numbers in numeric system are called orders. For polynomials was for the highest exponent set the term degree. Similarly the exponent of binomial equation has been called degree. In combinatorics the term class is used (e.g. the arrangement of n elements into k cells, contains nk variations of the k-th class). In the following we will try at least partially unify the terminology of designation of the exponent (order of the system, respectively degree of equation) to the letter k.

### Distribution of roots

Complex roots of the binomial equation xk−1 = 0 lies in the plane of complex numbers - spread out uniformly on the unit circle.

``` Exponent  Equation      Roots xj/k  for j=0,1,..,(k−1)
───────────────────────────────────────────────────────
1        x¹−1 = 0      xj/1 = cos 2π     + isin 2π
2        x²−1 = 0      xj/2 = cos 2π∙j/2 + isin 2π∙j/2
3        x³−1 = 0      xj/3 = cos 2π∙j/3 + isin 2π∙j/3
```

E.g. for x³−1=0 3 roots exists and makes equilateral triangle with vertices in points
1, −1/2+i√3/2 a −1/2−i√3/2
ie.:    x0/3= 1,   x1/3=−1/2+i√3/2,   x2/3=−1/2−i√3/2.

### The basic root

The zero root is x0/k=1. Basic root (or complex k-th root of the number 1) is α = x1/k. It makes an angle 2π/k with the x-axis on unit circle.

```  Exponent   Basic roots α  for exponents k=0,1,..,6
────────────────────────────────────────────────────────
1         α = cos 2π  + isin 2π      = +1
2         α = cos π   + isin π       = −1
3         α = cos 2π/3+ isin 2π/3    = −1/2+i√3/2
4         α = cos π/2 + isin π/2     = +i
...
6         α = cos π/3 + isin π/3     = +1/2+i√3/2
```

Powers of the root x1/k step through the whole circle ie. all are the powers of the basic root, i.e. every xj/k(k) is the j-th power of the root α:   xj/kj (for j=0,1,..,k−1)

For x4−1=0 there exists 4 roots and makes square with vertices in points 1,i,−1 a −i, i.e.:

```    x0/4= 1,
x1/4= i,
x2/4=−1 a
x3/4=−i.
```

These are so called complex units (see complex systems). Complex units are powers of the basic root α=x(1)=i:    α0= 1,   α1= i,    α²=−1  a  α³=−i.

### Integer analogy

For r=5,n=2 R-system R(2,4,5) of order 4 arise:

```   R(2,4,5)        Corrected R(2,4,5)    Complex units
0               0                    0
1  2  4  3      1  2  −1  −2         1  i  −1  −i
5               0                    0
```

Selt instances of system R(2,4,5) are an integer analogy of complex units, i.e. roots of binomial equation x4−1=0 with exponent k=r−1=4.
For α=n is accordint to module r:

```     α0=1,
α1=2,
α2=4 a
α3=3.
```
 Cayley, Arthur [keili], 1821-1895, English mathematician, founder of matrix theory, the theory of invariants and abstract theory of finite groups. He dealt with algebraic geometrie, studied n-dimensional spaces. He inspired use of combinatorics in structural chemistry.

Operations over finite sets of numbers can be clearly viewed using so called Cayley tables. All elements of the set are in the headers (vertical and horizontal) and the results of operations between elements of the headers are inside the table.

Tables for addition of numbers (additive tables) will be denoted T(r,+) tables for multiplication of numbers (multiplicative tables) T (r, ∙).

E.g. Cayley tables for adding and multiplying numbers in the field of congruence classes Z3:

```T(3,+)        T(3,*)
+ │ 0 1 2     * │ 0 1 2
──┼──────     ──┼──────
0 │ 0 1 2     0 │ 0 0 0
1 │ 1 2 0     1 │ 0 1 2
2 │ 2 0 1     2 │ 0 2 1
```

In the table T(3,+) is 2+1 (mod 3) = 0,

in the table T(3,∙) 2∙2 (mod 3) = 1.

The table, in which exists on every row and column a neutral element (unit) represents so called group.

```T(4,+)         T(4,*)
+ │ 0 1 2 3     * │ 0 1 2 3
──┼────────     ──┼────────
0 │ 0 1 2 3     0 │ 0 0 0 0
1 │ 1 2 3 0     1 │ 0 1 2 3
2 │ 2 3 0 1     2 │ 0 2 0 2
3 │ 3 0 1 2     3 │ 0 3 2 1
```

Residual classes Zr makes group with respect to aggregation, see table T(4 +).

In contrast, T(4 ∙) is not a group (unit is missing in line prefixed by number two).

Group G is a structure with one associative operation, whereby the inverse element x '(x ∙ x' = 1) exists to the each element x. Commutative group is a group with a commutative operation (i.e. it applies a∙b=b∙a for all elements a, b ε G). Commutative groups (also called Abel) were named acording to N.H. Abel (although he did not use exactly this concept).

Circuit O (4 + ∙) formed by tables T(4 +) and T(4 ∙) can not be field of integrity, because T(4 ∙) has no inverse element (see Groups). Circuit O (p, + ∙) makes of a body, when p is a prime number.

Any (non-empty) subset of elements of the group is called the complex. As with the elements, we can perform operations with the complexes. E.g. when a∙b=c, a∙c=d then a∙{b,c} = {c,d}.

 Borůvka, Otakar , 1899-1995, Czech mathematician. Built theory of groupoids and the theory of decompositions, founded the important algebraic school. He is known in graph theory by his algorithm for finding the minimum graph skeleton. Later he devoted to differential equations.

### Normal shape of tables

If all the units are on the main diagonal, the table is in normal form:

``` 1    1 a    1 a b     1 a b c
a 1    b 1 a     a 1 c b
a b 1     c b 1 a
b c a 1
```

There is only one reduced latin square of order 1, 2 and 3. Each of the two groups, which have 1,2 or 3 elements are therefore copies.

### Groups of solutions

Equation x−1 = 0
Trivial case with solution x=1 makes group of order 1:

```    R1     κ3
1      1
```

Equation x²−1 = 0
Equation has two solutions x1=1, 2=−1, i.e. (x−1)(x+1)=x²−1. It holds x1x1=x1, x1x2=x2x1=x2 a x2x2=x1, solutions creates (permutation) group of order 2. The arrangement corresponds to the roots of quadratic residues by module 5, group of parities of permutations, group of units in Z, etc.

```      │ x1 x2       R2   κ5     Parita    Jednotky
───┼─────────────────────────────────────────
x1 │ x1 x2       1 2    1 4     S L      1 −1
x2 │ x2 x1       2 1    4 1     L S     −1  1
```

Equation x³−1 = 0
Equation has three solutions x1=1, x2=(−1+i√3)/2, x3=(−1−i√3)/2. The same arrangement makes eg. group of quadratic residues according to module 7 (κ7, see Quadratic residues).
Groups of units 1,i,j established under relations i²=j, j²=i a i³=j³=ij=ji=1. Group of alignment operations of the equilateral triangle (ie. group of rotation through an angle I=0°,R=120°, S=240°).

```   │ x1 x2 x3   R3     κ7     Jednotky Z3+  Základní Zákryty
──┼────────  ──────────────────────────────────────────────
x1│ x1 x2 x3   1 2 3  1 2 4   1 i j   0 1 2   1 α α²   I R S
x2│ x2 x3 x1   2 3 1  2 4 1   i j 1   1 2 0   α α² 1   R S I
x3│ x3 x1 x2   3 1 2  4 1 2   j 1 i   2 0 1   α² 1 α   S I R
```

### Latin squares

The tables, which do not repeat in any row or column the same number are called Latin squares.

If numbers are sorted in natural order in the first row and column, the square is called reduced.

The number of Latin squares is larger than 1! 2! ... k!, wherein the reduced-squares Rk is always! (k-1)! times less [Bosak].

Number of reduced-squares Rk is for k<8:

```   R1  R2  R3   R4   R5    R6     R7
────────────────────────────────────
1   1   1   4    56   9408  16942080
```

Known are even some other values (R8,R9,...).

Group of solutions of binomical equations of order 1-3 corresponds to Latin squares R1-R3.

### Rotational group

Searching group of solutions for k = 4 - a first complication occurs. Latin squares of order 4 are four:

``` R4(a): 1234  R4(b): 1234  R4(c): 1234  R4(d): 1234
2143         2143         2341         2413
3412         3421         3412         3142
4321         4312         4123         4321
```

Three Latin squares R4(b,c,d) are mutually transferable by permutations of elements. Elements of these squares make up a single so called rotational group.

Equation x4−1=0, i.e.(x−1)(x+1)(x²+1)=0 has four solutions x1=1, x2=−1, x3=i a x3=−i. Strukture of solution corresponds to rotational group. Eg. these mathematical structures have a form of rotational groups:

· In the form R4(b) we recognize multiplication group of quadratic residues and non-residues (1,4,2,3) mod 5 and multiplication group of complex units:

```      │ x1 x2 x3 x4    R4(2)     K5         Jednotky
───┼───────────  ──────────────────────────────────
x1 │ x1 x2 x3 x4    1 2 3 4   1 4 2 3    1 −1  i −i
x2 │ x2 x1 x4 x3    2 1 4 3   4 1 3 2   −1  1 −i  i
x3 │ x3 x4 x2 x1    3 4 2 1   2 3 4 1    i −i −1  1
x4 │ x4 x3 x1 x2    4 3 1 2   3 2 1 4   −i  i  1 −1
```

· In the form R4(c) we find rotational group of square, additional group of numbers in Z4, group of (eclipsing) operations with square.

```  R4(3)      Z4           + │ I A B C    + │ I  R  R² R³
───────────────────     ──┼─────────   ──┼─────────────
1 2 3 4    0 1 2 3     I │ I A B C    I │ I  R  R² R³
2 3 4 1    1 2 3 0     A │ A B C I    R │ R  R² R³ I
3 4 1 2    2 3 0 1     B │ B C I A    R²│ R² R³ I  R
4 1 2 3    3 0 1 2     C │ C I A B    R³│ R³ I  R  R²
```
``` 1 2 3 4
2 4 1 3
3 1 4 2
4 3 2 1
```

For 4 possible slight rotations I,A=R, B=R², C=R³ I=0°, A=90°, B=180° and C=270° is operation of composition (eg. A+B = C, ie.R∙R²=R³) associative and also commutative. There is a neutral element (I) and for each element X there is an inverse element X', for which X+X'=I (eg. A+C=I)...

· In the form R4(d) we find the multiplication group of Z5. The same scheme is created by natural classes in G(2,5) (see Natural classes).

### Cyclic groups

Elements {1,a,a²,a³..,} make so called cyklic group. If the group is finite, a unitary element must be in succession 1,a,a²,..,ak−1,1, ... again. Exponent k is order of group. In group of order k, k-th power of any element makes unit of group (so called Fermat's theorem for groups).

Each subgroup of cyclic group is cyclic.

### Groups of the order 5

There are 56 Latin squares of order 5.

```   R5:  12345   12345   12345   12345  ...
21453   21453   21453   21453
34512   34521   35124   35214
45231   45132   43512   43521
53124   53214   54231   54132
```

All these squares are formed by permutations of basic elements.

Group of order 5 is formed eg. by residues according to module 11 (K11):

```  1 3 4 5 9
3 9 1 4 5
4 1 5 9 3
5 4 9 3 1
9 5 3 1 4
```

### Equation of the 5-th degree

Binomial equation x5−1=0 has five solutions, which in the plane of complex numbers form a pentagon (see the roots of the binomial equation).

Basic solution is:    x = cos 360°/5 + i∙sin 360/5° = cos 72° + i∙72°

Our goal is to get to this result through roots, without using trigonometric functions.

Equation M(x,5) = 0 has trivial root x=1, tedy (x4+x3+x2+x+1)(x−1) = 0.

From the first term after dividing by x² is: (x²+x+1+1/x+1/x²) = 0. If x+1/x=z then x²+1/x² = z²−2.

Substitution into the original equation gives z²+z−1=0 and from there two roots for z and four roots for x:

```  x0 =[1,0]
x1 =[(−1+√5)/4, +√2∙√(5+√5)/4]
x2 =[(−1−√5)/4, +√2∙√(5−√5)/4]
x3 =[(−1−√5)/4, −√2∙√(5−√5)/4]
x4 =[(−1+√5)/4, −√2∙√(5+√5)/4]
```
``` Check:
cos 72° = (-1+√5)/4
sin 72° = +√2*√(5+√5)/4
```

It holds:

·    when marking x1=x is:   g1 = x1+x4 = x1+x4 = [(−1+√5)/2,0]   g2 = x2+x3 = x2+x3 = [(−1−√5)/2,0]   i.e. g1+g2 = x1+x2+x3+x4 = x1+x2+x3+x4 = −1

·   roots x1,x4, resp. x2,x3, are symmetrical to the axis y.

·   roots x1,x2 are bound by goniometric relations: [cos 2x, sin 2x] = [cos² x− sin² x, 2∙sinx∙cosx].

 Lagrange, Josepf Louis, Comte[lagránž], 1736-1813, Italian-French mathematician and physicist, known for his contribution to the development of mechanics and astronomy. He was interested in the theory of numbers, algebraic and differential equations. He proposed a unified approach to the solution of algebraic equations until the fifth degree. He expressed the principle of least action in the form of an integral, which must have a minimum or maximum. He proved principle of virtual works and wrote D'Alembert's Principle analytically. He showed a special solution of the problem of three bodies.

## Nesting of roots

### Lagrange's theorem

Because (x−1)(x+1)=x²−1, solutions of euation x²−1 = 0 enter into the group of equation x4−1=0 Group of solutions of equatione x4−1=0 has therefore a subgroup of order 2.
Generally, the order of each subgroup G(d) divides order of group G(k). This is so called Lagrange's theorem.

But the reverse does not apply, there is not necessarily a subgroup G(d) to each divisor d.

The quotient k/d is called index of subgroup.

Order of subgroup d must divide order of group n, d | n. If n=pεP, there are only 2 subgroups - unit group and awn group G These subgroups are called trivial.

### Nesting of groups

Consider Cayley table T(r,∙), with non-prime modulus, e.g. r=6 (for completeness - we fill also number 0 to the table).

```M(6)
* │ 0 1 2 3 4 5
──┼────────────
0 │ 0 0 0 0 0 0
1 │ 0 1 2 3 4 5
2 │ 0 2 4 0 2 4
3 │ 0 3 0 3 0 3
4 │ 0 4 2 0 4 2
5 │ 0 5 4 3 2 1
```

Note that only two numbers: 0 and 3 occur for element R / 2 = 3 in the row and column of table T(6,∙). Three numbers 0,2 and 4 are in the rows and columns alongside.

```M(2):          M(3):
* │ 0 1       * │ 0 1 2
──┼────       ──┼──────
0 │ 0 0       0 │ 0 0 0
1 │ 0 1       1 │ 0 1 2
2 │ 0 2 1
```

It conveys the idea that these numbers do not belong to our table, but they were nested into it from tables of numbers belonging to divisors of number 6.

Indeed: {0,3} = (6/2)∙{0,1} a {0,2,4} = (6/3)∙{0,1,2},...

### Commutative groups of order 6

```R6:
123456 123456 ...
214365 214365
345612 345612
436521 436521
561234 561243
652143 652134
```

There are 9408 Latin squares of order 6.

Commutative group of quadratic residues and nonresidues

```  1 2 4  3 6 5     1    a    a²   b    ba   ba²
2 4 1  6 5 3     a    a²  1     ba   ba² b
4 1 2  5 3 6     a²  1    a     ba² b    ba
3 6 5  2 4 1     b    ba   ba²  a    a²  1
```

Operation f(a,b) = a∙b (mod 7), for a=2, b=3.

``` 1 2 4  3 6 5       1  3  4   9 10 12
2 3 5  1 4 6       3  9 12   1  4 10
4 5 2  6 1 3       4 12  3  10  1  9
3 1 6  2 5 4       9  1 10   3 12  4
6 4 1  5 3 2      10  4  1  12  9  3
5 6 3  4 2 1      12 10  9   4  3  1
```

### Non-commutative group of permutations

Eclipsing operations in triangle, D1,D2,D3 - overturn according to the axis of side (angle), R - rotation. It holds: D1²=D2²=D3²=1 R³=1.

```  1  R  R²   D1 D2  D3
R  R² 1    D3 D1  D2
R² 1  R    D2 D3  D1
D1 D2 D3    1  R  R²
D2 D3 D1    R² 1  R
D3 D1 D2    R  R² 1
```
``` 1 2 4   3 6 5    1   a   a²    b   ba  ba2
2 4 1   6 5 3    a   a²  1     ba  ba² b
4 1 2   5 3 6    a²  1   a     ba² b   ba
3 5 6   1 4 2    b   ba² ba    1   a²  a
6 3 5   2 1 4    ba  b   ba²   a   1   a²
5 6 3   4 2 1    ba² ba  b     a²  a   1
```

For aε K7 is f(a,b) = a∙b  (the upper half of the table);  resp. f(a,b) = a/b (the lower half of the table).

It holds a³=b²=1, abab= 1 (i.e. aba=b, ab=b/a, ab = ba²) .

### Generators of cyclic group

```Vlastní čísla T(6,*):
* │ 0 1 2 3 4 5
──┼────────────
0 │ - - - - - -
1 │ - 1 - - - 5
2 │ - - - - - -
3 │ - - - - - -
4 │ - - - - - -
5 │ - 5 - - - 1
```

Generators of additive cyclic group Zn may be just numbers relatively prime to n.

Each table T(r,∙) has always only φ(r) self rows and columns, the other rows and columns are nested from the tables with orders d(i)..., where d(i) | r (number i runs through divisors of number r).

Whereas group (Z5,+) can be generated by any element {1,2,3,4}), groupu (Z6,+) can be generated by element 1 or 5.

Cyclical group of order r has φ(r) generators. Cyclical group of order r has as many subgroups as there are (positive) divisors of number r. When r=pk ...

### Orders of roots

Order of root xj is the smallest number s, that: xjs−1 = 0. For basic root is s=k (exponent of binomial equation). Some of the other roots may have a lower order. When k=6 for roots x0,..,x5 is:     x0 = x16 = x2³ = x3² = x4³ = x56

Hence orders (s) of roots xj(6):         Index of root j  0  1  2  3  4  5    Order of root s  1  6  3  2  3  6

Generally k | j∙s, ie. exponent of binomial equation is divisor of product of root number and its order (see bases belonging to orders.)

### Primitive roots

Denote Mh(x) product of all roots and Vh(x) product of all primitie roots of binomical equation xh−1 = 0. The roots represent "all instances", primitive roots "self instances" (as defined in paragraphs about G-systems).

```  All instances   Self instances (for h=1..6)
────────────────────────────────────────────────
M1(x)=x−1      V1(x)=x−α1 = x−1
M2(x)=x²−1     V2(x)=x−α2 = x+1
M3(x)=x³−1     V3(x)=(x−α3)(x−α3²) =[x−(−1+i√3)/2]∙[x−(−1−i√3)/2]=x²+x+1
M4(x)=x4−1     V4(x)=(x−i)(x−i³) =(x−i)∙(x+i) = x²+1
M6(x)=x6−1     V6(x)=(x−α6)(x−α65)=[x−(1+i√3)/2]∙[x−(1−i√3)/2]=x²−x+1
```

Relations V2(x) = x+1=(x²−1)/(x−1), V3(x) = x²+x+1=(x³−1)/(x−1) seem to suggest that for pεP is number of "self instances": Vp(x) =(xp−1)/(x−1)

### Principle of nesting

If pεP then instances are nested into Mp(x) = xp−1 only from the system of order 1. We eliminate these nested instances by dividing by expression V1(x)=(x−1). Therefore Vp(x) = Mp(x)/V1(x) = (xp−1)/(x−1).

We similarly get self instances for non-primes h - by exclusion of all nested systems Md(x) d | h. From relations   (x4−1) = (x²−1)(x²+1) = (x−1)(x+1)(x²+1)   (x6−1) = (x³−1)(x³+1) = (x−1)(x²+x+1)(x+1)(x²−x+1)

it follows:
```    V4(x) = M4(x)/V1(x)/V2(x)=(x4−1)/(x−1)/(x+1)= x²+1
V6(x) = M6(x)/V1(x)/V2(x)/V3(x)
=(x6−1)/(x−1)/(x+1)/(x²+x+1)=x²−x+1
```

Generally:

### Groups of the order 8

From elements C(2,4) 2 different non-commutative acyclic groups of order 8 arise: The first algebra for b² = 1 is called diedric and is denoted D8, the second, so called quaternion, satisfies a² = b² and is denoted Q8 [Procházka].

```Kvaternion group
b² = a², ab= ba³
(example for a=2,b=3, mod 15)
1 2 4 8 3 6 12 9
2 4 8 1 9 3 6 12
4 8 1 2 12 9 3 6
8 1 2 4 6 12 9 3
3 6 12 9 4 8 1 2
9 3 6 12 2 4 8 1
12 9 3 6 1 2 4 8
6 12 9 3 8 1 2 4
```
``` Diedric group
b² = 1 , ab= ba³
(example for a=2,b=3, mod 15)
1 2 4 8 3 6 12 9
2 4 8 1 9 3 6 12
4 8 1 2 12 9 3 6
8 1 2 4 6 12 9 3
3 6 12 9 1 2 4 8
9 3 6 12 8 1 2 4
12 9 3 6 4 8 1 2
6 12 9 3 2 4 8 1
```

Alternative (non-associative, non-commutative) algebra is called Cayley algebra.

It applies: i²=j²=k²=e²=−1

```   1   i   j   k  │  e  ie  is   ke
i  −1   k  −j  │ ie  −e  −ke  is
j  −k  −1   i  │ is  ke  −e  −ie
k   j  −i  −1  │ ke  −is ie  −e
───────────────┼─────────────────
e  −ie −is −ke │ −1   i   j   k
ie  e  −ke  is │ −i  −1  −k   j
is  ke  e  −ie │ −j   k  −1  −i
ke −is  ie  e  │ −k  −j   i  −1
```

Group for multiplication of quaternions (upper left quadrant) is a subgroup of Cayley algebra.