%0 Journal Article
%T Application of the Todd-Coxeter Algorithm in the Computation of Group Theory
%A Moumouni Djassibo Woba
%J Advances in Linear Algebra & Matrix Theory
%P 37-52
%@ 2165-3348
%D 2023
%I Scientific Research Publishing
%R 10.4236/alamt.2023.133003
%X 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>i.e.</i> different ways of expressing its elements and operations. We have also applied this algorithm to a subgroup generated <i>H</i> by <i>G</i>; where we obtained a table of the subgroup, three tables of relators including: Table of the relator aaaa; Table of the relator <i>abab</i>; Table of the relator <i>bbb</i> and a multiplication table <i>aa'bb</i><i>'</i>. Once the algorithm is complete, the unit of <i>H</i> in <i>G</i> is 6. We have explicitly obtained a homomorphism of <i>G</i> in the group of permutations of <i>H</i>/<i>G</i> which is isomorphic to <i>G</i><sub>6</sub>; where we have noticed that it is injective: in fact, an element of the nucleus belongs to the intersection of the <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>x</mi><mi>H</mi><msup>
   <mi>x</mi>
   <mrow>
    <mo>&#x2212;</mo><mn>1</mn></mrow>
  </msup>
  </mrow>
</math>

 for <math display='inline' xmlns='http://www.w3.org/1998/Math/MathML'>
 <mrow>
  <mi>x</mi><mo>&#x2208;</mo><mi>G</mi></mrow>
</math>

, in particular, it belongs to <i>H</i>; on the other hand, the image of <i>H</i> in <i>G</i><sub>6</sub> is of order 4, so the nucleus is reduced to the neutral element.
%K Todd-Coxeter Algorithm
%K Subgroup
%K Semi-Direct
%K Operating Group
%K Homomorphism
%U http://www.scirp.org/journal/PaperInformation.aspx?PaperID=136845