We denote ${\mathfrak{G}}_n$ the symmetric group of degree n, i.e. the group of bijections of the set with n elements $\left\{1,\cdots ,n\right\}$ in itself equipped with the law of composition. A permutation of degree n is an element of ${\mathfrak{G}}_n$. A group of permutations (of degree n) is a subgroup of ${\mathfrak{G}}_n$. Recall that there is a single homomorphism of ${\mathfrak{G}}_n\to \left\{\pm 1\right\}$ such that the image of a transposition is −1. It’s the signature. The nucleus is the alternating group 𝔘_n.

U_n r-cycle cycle (or r-length cycle) is denoted $\left(a_1{a}_2\cdots a_r\right)$. This is the permutation that sends $a_1$ to $a_2\cdots a_r$ to $a_1$. The set $\left(a_1\cdots a_r\right)$ is called the support of the cycle $\left(a_1{a}_2\cdots a_r\right)$.

The group ${\mathfrak{G}}_n$ is generated by (1 2) and (1 2...n). Group ${\mathfrak{U}}_n$ is generated for $n>3$ by (1 2 3) and (3...n) if n is odd and by (1 2 3) and (1 2) (3...n) if n is even.

The first statement is classical. For the second, we recall that ${\mathfrak{U}}_n$ is generated by the 3-cycles (1 2 i) for $i\in \left\{3\cdots n\right\}$. If $c=\left(1,2,3\right)$ and $\sigma =\left(1,2\right)\left(3\cdots n\right)$ or $\sigma =\left(3\cdots n\right)$ according to the parity of n, we have ${\sigma }_{i}c{\sigma }_{-1}$ = (1 2 3 + i) or (2 1 3 + i) = (1 2 3 + i)². The preceding proposition is easily deduced from this.

Let X be a set. A group G operating (left) on X is a group with an application $G\times X\to X:\left(g,x\right)\mapsto g\cdot x$ verifying $\left(g{g}^{\prime }\right)\cdot x=g\cdot \left(g^{\prime }x\right)$ for any $g,g^{\prime }\in G$ and $x\in X$ and $e\cdot x=x$ if e is a neutral element of G. It is the same thing to give oneself a homomorphism of groups $G\to S\left(X\right)$ where $S\left(X\right)$ designates the group of permutations of X.

G is said to operate transitively on X if for all x and $y\in X$, there exists $g\in G$ such that $gx=y$. It is the same thing to say that the orbit of any $x\in X$, that is, the set of $g\cdot x$ for $g\in G$, is equal to X. If G is defined as a group of permutations of degree n and its natural action on $\left\{1,\cdots ,n\right\}$ is transitive, G is said to be transitive.

G is said to operate 2-transitively on X if for all $\left(x,y\right)\in X\times X$ with $x\ne y$ and for $\left(x^{\prime },y^{\prime }\right)\in X\times X$ with $x\ne y$, there exists $g\in G$ such that g $x=x^{\prime }$ and $gy=y^{\prime }$. In particular, G operates transitively on X and G operates 2-transitively if and only if the diagonal action of G on $X\times X:g\cdot \left(x,y\right)=\left(g\cdot x,g\cdot y\right)$ has exactly two orbits: the diagonal of $X\times X$ and the complement. A group of permutations of degree n operating 2-transitively on $\left\{1,\cdots ,n\right\}$ is said to be 2-transitive [3] .

Two elements of 𝔊_n are conjugated if and only if their decompositions into disjoint cycles have for all i the same number of cycles of length i.

A group G operates on itself by translation (left): $\left(g,x\right)\mapsto g\cdot x$. If H is a subgroup of G, the group G also operates on the set G/H by translations, which allows us to define a homomorphism of ρ groups: $G\to S\left(G/H\right)$ by $\rho \left(g\right)\left(C\right)=gC$ for $g\in G$ and C an element of G/H. In particular, once these Additional definitions: derived subgroup, semi-direct product are numbered from 1 to n if n is the index of H in G, we obtain a homomorphism ρ' of G in 𝔊_n. Note that ${\rho }^{\prime }\left(G\right)$ is a group of transitive permutations of degree n, quotient of G.

Let G be a group. A commutator is an element d G of the form $xy{x}^{-1}{y}^{-1}$. We call the derivative group of G (and we denote G' or D(G)) the subgroup generated by the switches of G.

D(G) is a distinguished subgroup of G. The quotient G/D(G) is an abelian group and even the largest abelian quotient group of G in the following sense: let us $\pi :G\to G/D\left(G\right)$; if G₁ is an abelian group and $f:G\to G_1$ a homomorphism of groups, there exists a unique homomorphism $\bar{f}:G/D\left(G\right)\to G_1$ such that $\bar{f}\circ \pi =f$. Thus, the order of G/D(G) is maximal among the order of the quotients of G that are abelian. A sequence derived from a group G is called the sequence $G_0=G,G_1=D\left(G_0\right),\cdots ,G_k=D\left(G_{k-1}\right),\cdots$

The group derived from 𝔊_n is ${\mathfrak{U}}_n$. The group derived from ${\mathfrak{U}}_n$ is ${\mathfrak{U}}_n$ for $n\ge 5$. for $n=4$, ask MAPLE later for their thoughts [4] .

Let H and K be two groups and $T:K\to Aut\left(H\right)$ a group homomorphism (here, Auto(H) is the group of bijective homomorphisms of H in itself). The semi-direct (abstract) product of H by K with respect to T is the set $H\times K$ with the following law.

$\left(h,k\right)\ast \left(h^{\prime },k^{\prime }\right)=\left(hT\left(k\right)\left(h^{\prime }\right),k{k}^{\prime }\right).$

This law is a group law, let us denote G group. When T is the trivial homomorphism, we find the direct product. It is easy to show that the sets $H^{\prime }=H\times \left\{1\right\}$ and $K^{\prime }=\left\{1\right\}\times K$ are the subgroups of G, that H' is distinguished in G and that $K^{\prime }\cap H^{\prime }=\left\{1\right\}$.

Let G be a group and let H and K be two subgroups of G. G is said to be the semi-straight product of H by K if H is distinguished in G, if $G=HK$ and if $H\cap G=\left\{1\right\}$.

Under the previous conditions, let us take $T:K\to Aut\left(H\right)$ gives by $k\mapsto \left(h\mapsto kh{k}^{-1}\right)$. Then G is isomorphic to the semidirect (abstract) product of H by K with respect to T.

Let V be a set. The free monoid of base V is the set note $V^{*}$, of finite sequences of elements of V. These sequences are denoted by juxtaposing the elements, for example the sequence $\left({\upsilon }_1,\cdots ,{\upsilon }_r\right)$ with ${\upsilon }_i\in V$ is denoted ${\upsilon }_1,\cdots ,{\upsilon }_r$. The elements of $V^{*}$, are called strings of elements of V or words on V.

If $\upsilon ={\upsilon }_1,\cdots ,{\upsilon }_r\in V^{*}$, with ${\upsilon }_i\in V$, The length of the string $\upsilon$ is r. We denote ρ the natural map $V\to V^{*}$ which to $\upsilon$ associates the chain of length 1 formed by $\upsilon$. If ${\upsilon }_1$ and ${\upsilon }_2$ are words, the word formed by juxtaposing them is denoted ${\upsilon }_1{\upsilon }_2$ and is called the concatenation of ${\upsilon }_1$ and of ${\upsilon }_2$.

The concatenation defines an internal composition law on $V^{*}$ which is associative and admits a neutral element ε which is the empty string.

For any monoid M and any map $f:V\to M$, there exists a single monoid morphism $f^{*}:V^{*}\to M$ such that $f=f^{*}o\rho$.

Note that $V^{*}$ is characterized by the preceding universal property with a single isomorphism. Let V' be a copy of V. If $\upsilon$ is an element of V, we denote ${\upsilon }^{\prime }$ the same element seen in V'.

Let $\bar{V}=V\bigsqcup V^{\prime }$ be the disjoint meeting of V and V'. Let ${\bar{V}}^{*}$ be the free monoid of base V'. Si $\alpha ={\alpha }_1\cdots {\alpha }_r$, we set ${\alpha }^{\prime }={\alpha }_{r^{\prime }}\cdots {\alpha }_{1^{\prime }}$.

Two elements of ${\bar{V}}^{*}$ are said to be equivalent if there are $n\in \mathbb{N}$ and ${\alpha }_0,{\alpha }_1,\cdots ,{\alpha }_n$ belong to ${\bar{V}}^{*}$ such that ${\alpha }_0={\alpha }_1\cdot {\alpha }_n=\beta$ and such that if $0\le i<n$, ${\alpha }_i$ and ${\alpha }_{i+1}$ are contiguous. The relation thus defined is an equivalence relation. The parity of the length is conserved by this relation.

The free group F(V) of base V is the set of equivalence classes of the previous equivalence relation. We check that the concatenation respects the equivalence relation. This makes it possible to define an internal composition law on F(V).

Equipped with the concatenation, F(V) is a group. For $\alpha \in V$, the inverse of (the class) of $\alpha$ is (The class of) ${\alpha }_0$. For this reason, ${\alpha }_0={\alpha }^{-1}$ for $\alpha \in V$ is also denoted. We still have a map $\bar{\rho }:V\to F\left(V\right)$.

For any group G and any map $f:V\to G$, there exists a unique homomorphism of group $\bar{f}:F\left(V\right)\to G$ such that $f=\bar{f}o\rho$. Again, F(V) with the map $\bar{\rho }:V\to F\left(V\right)$ is characterized with a single isomorphism by the universal property above.

1) If V is the empty set, the base free group V is the trivial group {1}.

2) If $V=\left\{\alpha \right\}$ is reduced to one element, F(V) is isomorphic to $\mathbb{Z}$ Indeed, we begin to enumerate the elements of F(V) by “reducing” them: these are the α...α (n times ) = αⁿ and the α'...α' (n times) = α'ⁿ = α⁻ⁿ for $n\in \mathbb{N}$. Notice that $\mathbb{Z}$ verifies the universal property: if G is a group, an application.

$f:\left\{\alpha \right\}\to G$ is determined by the image $b=f\left(\alpha \right)\in G$; there is then a single group homomorphism of $\mathbb{Z}$ in G given by $n\to b^n$. By uniqueness of the universal object, we deduce that F(V) is isomorphic to $\mathbb{Z}$.

3) If V has two elements α and β, F(V) is very large: it contains, for example αb, αbα, αbα⁻¹b, α⁵b²αbαbαb, etc. which are all distinct.

Let be a group and A, a part of G. The distinguished subgroup of G generated by A is the intersection of all the distinguished subgroups of G containing A. It is also the smallest distinguished subgroup of G containing A. It is formed of the finite products of elements of A and all their conjugates by an element of G.

A group presentation is a pair (X, R) where X is a set and R is a part of F(X). Let $G=\langle X/R\rangle$ be the quotient of the free group F(X) by the distinguished subgroup of F(X) generated by R. We say that (X, R) is a presentation of G or that it is a definition of G by generators and relations. The elements of X are called generators, the elements of R are called relators. If r is a relator, r = 1 is called a relation. We also denote $\langle X/R\rangle =\langle X|\omega =1\text{for}\omega \in R\rangle$.

By example

$\langle \mathcal{x},\mathcal{y}|x^2,y^2,\mathcal{x}\mathcal{y}{\mathcal{x}}^{-1}{y}^{-1}\rangle$ or $\langle \mathcal{x},\mathcal{y}|x^2=1,y^2=1,\mathcal{x}\mathcal{y}=\mathcal{y}\mathcal{x}\rangle$,

$\langle \mathcal{x},\mathcal{y}|x^n,y^2,\mathcal{y}\mathcal{x}{\mathcal{y}}^{-1}\mathcal{x}\rangle$ or $\langle \mathcal{x},\mathcal{y}|x^n=1,y^2=1,\mathcal{y}\mathcal{x}{\mathcal{y}}^{-1}={\mathcal{x}}^{-1}\rangle$.

For any group G and any map $f:X\to G$ such that $\bar{f}\left(r\right)=1$ for $r\in R$, there exists a unique homomorphism of group $f^{\prime }:\langle X/R\rangle \to G$ such that $f=f^{\prime }o\pi o\bar{\rho }$ where π is the projection of F(X) onto $\langle X/R\rangle$.

1) If R is the empty set, the subgroup of F(X) generated by R is reduced to 1. So $\langle H/R\rangle =F\left(X\right)$.

2) $G=\langle a/a^5\rangle$ 〉 is isomorphic to $\mathbb{Z}/5\mathbb{Z}$. Indeed, we have $F\left(\left\{a\right\}\right)=a^{\mathbb{Z}}$, the distinguished subgroup generated by $a^5$ is $a^{5\mathbb{Z}}$.

3) $G=\langle a,b/a^3,b^2,ab{a}^{-1}{b}^{-1}\rangle$ is isomorphic to $\mathbb{Z}/6\mathbb{Z}$. Indeed, in ${\bar{X}}^{*}$ $ab~ab{a}^{-1}a~ab{a}^{-1}{b}^{-1}ba~ba$. We enumerate the elements of G: we find 1, $a,a^2,b,ba,b{a}^2$. So $\left|G\right|\le 6$. On the other hand, the group $\mathbb{Z}/6\mathbb{Z}$ has two generators x = 2 and y = 3 verifying (additive notation) $\mathcal{x}=0$, $2\mathcal{y}=0$, $\mathcal{x}+\mathcal{y}-\mathcal{x}-\mathcal{y}=0$. We deduce from this by the universal property that there exists a group homomorphism $G\to \mathbb{Z}/6\mathbb{Z}$ which sends a over x = 2 and b over y = 3. It is surjective. Hence $\left|G\right|\ge 6$. So $\left|G\right|=6$ and G is isomorphic to Z/6Z.

4) Recognize the groups $\langle \mathcal{x}/{\mathcal{x}}^n=1\rangle$, $\langle \mathcal{x},\mathcal{y}|x^2=1,y^2=1,\mathcal{x}\mathcal{y}=\mathcal{y}\mathcal{x}\rangle$.

The Todd-Coxeter algorithm is best known for solving problems related to group theory, by allowing finite presentations of groups to be calculated. However, this algorithm also has interesting applications in other fields:

1) Combinatorial optimization: The Todd-Coxeter algorithm can be used to solve some combinatorial optimization problems, such as the traveling salesman problem. By modeling the problem in the form of relationships between cities (such as generators and relationships in a group), we can use the algorithm to find optimal solutions.

2) Formal language theory: The algorithm can be applied to the study of formal languages, in particular to compute finite automata equivalent to algebraic grammars. This makes it possible to obtain compact and efficient representations of certain languages.

3) Cryptography: In some cryptographic schemes based on group theory, the Todd-Coxeter algorithm can be used to efficiently manipulate the representations of the groups involved.

4) Algebraic topology: In knot theory, for example, the algorithm can help compute topological invariants by modeling knots as groups of braids.

5) Mathematical physics: Some mathematical models in physics, such as crystal lattices, can be studied using the Todd-Coxeter algorithm to understand their algebraic properties.

6) Computer science theory: the algorithm has links to classical problems of complexity theory, such as the group isomorphism problem.

Although the Todd-Coxeter algorithm is historically associated with group theory, these examples show that it can be useful in many other fields requiring the efficient computation of finite algebraic structures. Its adaptability makes it a powerful tool for solving a wide variety of practical problems.

Let G be a group defined by generators and relations: $G=\langle X|R\rangle$ with finite X and H a subgroup of G generated by the image of a finite subset S of words of $\bar{X}=X\bigsqcup X$. The Todd-Coxeter algorithm that we are going to describe allows, when the index of H in G is finite, to calculate this index and to give the action of G by right translation on the set of classes HG.

It is a question of giving all classes a number and only one, class H having for example the number 1 (not to be confused with the neutral element). To do this, we will build a certain number of arrays according to the following rules (we recommend doing the following example at the same time).

I) The key steps and principles of the Todd-Coxeter algorithm, an important algorithm in group theory:

a) Group performances:

The algorithm starts by representing the group using generators. This can be done in the form of a group presentation.

b) Initialization: The algorithm starts with an initial set of side classes, which represent the elements of the group. Often, we start with a single class, corresponding to the neutral element of the group.

c) Relationship Processing: At each step, the algorithm takes a relationship between generators and tries to apply it to existing side classes. This can lead to some side classes being identified as identical, thus reducing the total number of classes.

d) Iterative process: the algorithm proceeds iteratively, successively applying the group’s relationships to the lateral classes, until no new identification is possible.

e) Termination: The algorithm terminates when the application of the relationships no longer results in any new identification of side classes. At this point, the final set of side classes represents the elements of the group.

f) Key Principles:

NB. The Todd-Coxeter algorithm is very useful for studying group structure and calculating invariants such as group order. It has many applications in group theory, algebraic topology, and other mathematical fields.

1) For each word $\alpha =a_{i1}\cdots a_{ir}$ of S, we associate a table M_gen(α) with a single row and r + 1 columns whose first element is 1. We will fill in the second column later by putting the number of the class of $H_{a_{i1}}$ which we also note 1. $a_{i1}$, then the third column with the number of 1. $a_{i1}\cdot a_{i2}$, etc.

First principle: The last element in the line is 1. Indeed, if $\alpha \in S$, we have $H\alpha =H$. These tables are called the tables of subgroup H. The tables are presented here in the initial state:

The Todd-Coxeter algorithm uses this relative table to systematically explore all the elements of the group and identify its subgroups. It proceeds as follows:

1) We start with a set of generators of the group and their relationships.

2) The relative table is constructed by filling each cell with the result of multiplying a row item by a column item, according to the relationships of the group.

3) The algorithm explores the relative table in a systematic way, identifying the items in the group and inferring the subgroup structure.

The multiplication table is another key tool. It is deducted from the relative table and represents the products of all the items in the group. Each box contains the result of multiplying the row item by the column item.

Analysis of the multiplication table makes it possible to identify the subgroups of the initial group. Indeed, the subgroups correspond to blocks of closed squares in the multiplication table, i.e. areas where the multiplications remain inside.

In summary, the Todd-Coxeter table, through its relative table and its multiplication table, offers a compact and powerful representation of the algebraic structure of a group. The Todd-Coxeter algorithm exploits this representation to systematically explore subgroups, which is essential for understanding the overall structure of the group.

Let’s take $G=\langle a,b|a^4={\left(ab\right)}^2=b^3=1\rangle$.

So we have $X=\left\{a,b\right\}$, $R=\left\{aaa,abab,bbb\right\}$

Let us take for H the subgroup generated by a. So there is a subgroup table, three relator tables and a multiplication table [6] .

At first, they are of the following form:

Subgroup table:

We have explicitly obtained a homomorphism of G in the group of permutations of HG which is isomorphic to 𝔊₆. Note that it is injective: in fact, an element of the nucleus belongs to the intersection of $\mathcal{x}H{\mathcal{x}}^{-1}$ for $\mathcal{x}\in G$, in particular, it belongs to H; on the other hand, the image of H in 𝔊₆ is of order 4, so the nucleus is reduced to the neutral element.