Symmetric Generatic Generations and an Algorithm to Prove Relations

Abstract

When researching this thesis, we found that various progenitors, including 3^*56:(2³:(3:7), Factoring m*: N by the First Order Relations in Permutation Progenitors, and Symmetric Generation, each exhibit homomorphic pictures. To generate the Cayley graph of the ratio 3:(2×S₅) over D₁₂, we manually do Double Coset Enumeration. When dealing with finitely presented groups, coset enumeration is the primary technique for addressing the word problem. This method has been around for quite some time, and it was really one of the earliest uses of electronic computers for purely mathematical purposes. The current programs' functionality is limited by the size of the coset table that can be stored in the computer's memory. In certain situations, a significant space savings may be realized by the enumeration of double cosets. There is a description of an algorithm followed by some implementation details. We also show some of the isomorphism types and original symmetric presentations of finite groups, and we explain them as homomorphic pictures of their progenitors.