Decoding Groups: Innovative applications of the Todd-Coxeter algorithm
Keywords:
group theory, relators, algebraic structures, isomorphism, permutation groupAbstract
In this article, we delve into the Todd-Coxeter algorithm, an influential mathematical tool in the realm of group theory. This algorithm empowers us to uncover various presentations of a group, showcasing diverse methods for representing its elements and operations.
We applied the algorithm to a subgroup ???? generated by ????, resulting in a detailed subgroup table, along with three relator tables for ????????????????, ????????????????, and ????????????, as well as a multiplication table. The completion of our analysis revealed the unit of ???? in ???? to be 6.
Moreover, we established a clear homomorphism from ???? to the permutation group of, which is isomorphic to ????6. Significantly, this homomorphism is injective, indicating that the elements of the nucleus intersect with ????. Additionally, the image of ???? in ????6 has an order of 4, confirming that the nucleus reduces to the identity element.
