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

作　　者：Moumouni Djassibo Woba Moumouni Djassibo Woba(Unit de Formation et de Recherhe, Universit de Ouahigouya, Ouahigouya, Burkina Faso)

机构地区：[1]Unit de Formation et de Recherhe, Universit de Ouahigouya, Ouahigouya, Burkina Faso

出　　处：《Advances in Linear Algebra & Matrix Theory》2023年第3期37-52,共16页线性代数与矩阵理论研究进展（英文）

摘　　要：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 xHx−1for 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.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 xHx−1for 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.

关 键 词：Todd-Coxeter Algorithm SUBGROUP Semi-Direct Operating Group HOMOMORPHISM 

分 类 号：O18[理学—数学]