ApCoCoA:Mathieu11 group

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Description

The Mathieu group M11 is a sporadic group with order 7920 and it is a subgroup of the monstergroup.

Reference

ATLAS of Finite Group Representations - Version 3

Computation

 /*Use the ApCoCoA package ncpoly.*/
   
  Use ZZ/(2)[a,b];
  NC.SetOrdering("LLEX");
  
   Define CreateRelationsMathieuM11()
    Relations:=[];
    
    // add the relation a^2 = 1
    Append(Relations,[[a,a],[1]]);
    // add the relation b^4 = 1
    Append(Relations,[[b,b,b,b],[1]]);
 
    //add the relation (ab)^11 = 1
    Append(Relations,  [[a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b,a,b],[1]]);
   
    //add the relation (ab^2)^6 = 1
    Append(Relations,[[a,b,b,a,b,b,a,b,b,a,b,b,a,b,b,a,b,b],[1]]);
    
    //add the relation ababab^{-1}abab^2ab^{-1}abab^{-1}ab^{-1} = 1
    Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,b,a,b,b,b,a,b,a,b,b,b,a,b,b,b],[1]]);
    
    Return Relations;
  EndDefine;
  
  Relations:=CreateRelationsMathieuM11();
  Relations;
  
  Gb:=NC.GB(Relations,31,1,100,1000);
  Gb;

Example in Symbolic Data Format

 <FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier">
 	<vars>a,b</vars>
 	<uptoDeg>27</uptoDeg>
 	<basis>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>b*b*b*b-1</ncpoly>
 	<ncpoly>(a*b)^11-1</ncpoly>
 	<ncpoly>(a*b*b)^6-1</ncpoly>
 	<ncpoly>a*b*a*b*a*b*b*b*a*b*a*b*b*a*b*b*b*a*b*a*b*b*b*a*b*b*b-1</ncpoly>
 	</basis>
 	<Comment>The partial LLex Gb has 15 elements</Comment>
 	<Comment>Mathieu group_M_{11}</Comment>
 </FREEALGEBRA>