ApCoCoA-1:Mathieu22 group

Mathieu Group M {22}

Description

The Mathieu group M22 is a sporadic group with order 443520 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,c];
  NC.SetOrdering("LLEX");
  
   Define CreateRelationsMathieuM22()
    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)^5 = 1
    Append(Relations,[[a,b,b,a,b,b,a,b,b,a,b,b,a,b,b],[1]]);
    
    //add the relation [a,bab]^3=1
    // the commutator of [a,bab] is a,b,a,b,b^3,a,b^3
    Append(Relations,[[a,b,a,b,b,b,b,a,b,b,b,a,b,a,b,b,b,b,a,b,b,b,a,b,a,b,b,b,b,a,b,b,b],[1]]);
 
    //add the relation (ababab^-1)^5 = 1
    //Append(Relations,[[a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b,a,b,a,b,a,b,b,b],[1]]);
    Append(Relations,[[a,b,a,b,a,b,b,b],[c]]);
    Append(Relations,[[c,c,c,c,c],[1]]);
    Return Relations;
  EndDefine;
  
  Relations:=CreateRelationsMathieuM22();
  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>12</uptoDeg>
 	<basis>
 	<ncpoly>a*a-1</ncpoly>
 	<ncpoly>b*b*b*b-1</ncpoly>
 	<ncpoly>((a*b)^11)-1</ncpoly>
 	<Comment>relation: (ab)^11=1 </Comment>
 	<ncpoly>(a*b*b)^5-1</ncpoly>
 	<Comment>relation: (abb)^5=1</Comment>
 	<ncpoly>(a*b*a*b*b*b*b*a*b*b*b)^3-1</ncpoly>
 	<Comment>commutator: [a,bab]=ababb^3ab^3</Comment>
 	<Comment>relation [a,bab]^3=1</Comment>
 	<ncpoly>(a*b*a*b*a*b*b*b)^5-1</ncpoly>
 	<Comment>relation (ababab^-1)^5=1</Comment>
 	</basis>
 	<Comment>The partial LLex Gb has 186 elements</Comment>
 	<Comment>Mathieu group_M_{22}</Comment>
 </FREEALGEBRA>