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 presenta...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.展开更多
文摘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.
