TITLE:
Application of the Todd-Coxeter Algorithm in the Computation of Group Theory
AUTHORS:
Moumouni Djassibo Woba
KEYWORDS:
Todd-Coxeter Algorithm, Subgroup, Semi-Direct, Operating Group, Homomorphism
JOURNAL NAME:
Advances in Linear Algebra & Matrix Theory,
Vol.13 No.3,
October
25,
2024
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 G6; where we have noticed that it is injective: in fact, an element of the nucleus belongs to the intersection of the
xH
x
−1
for
x∈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.