Application of the Todd-Coxeter Algorithm in the Computation of Group Theory

Abstract

In this article, we have described the Todd-Coxeter algorithm. Indeed, the Todd-Coxeter algorithm is a mathematical tool used in the field of group theory. It makes it possible to determine different possible presentations of a group, i.e. different ways of expressing its elements and operations. We have also applied this algorithm to a subgroup generated H by G ; where we obtained a table of the subgroup, three tables of relators including: Table of the relator aaaa; Table of the relator abab ; Table of the relator bbb and a multiplication table aa'bb ' . Once the algorithm is complete, the unit of H in G is 6. We have explicitly obtained a homomorphism of G in the group of permutations of H / G which is isomorphic to G 6 ; where we have noticed that it is injective: in fact, an element of the nucleus belongs to the intersection of the x H x − 1 for x ∈ G , in particular, it belongs to H ; on the other hand, the image of H in G 6 is of order 4, so the nucleus is reduced to the neutral element.