Difference between revisions of "ApCoCoA-1:Mathieu22 group"
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StrohmeierB (talk | contribs) (New page: === <div id="Mathieu22_group">Mathieu group M {22}</div> === ==== Description ==== The Mathieu group M22 is a sporadic group with order 443520 an...) |
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| − | === <div id="Mathieu22_group">[[:ApCoCoA:Symbolic data#Mathieu22_group|Mathieu | + | === <div id="Mathieu22_group">[[:ApCoCoA:Symbolic data#Mathieu22_group|Mathieu Group M {22}]]</div> === |
==== Description ==== | ==== Description ==== | ||
The Mathieu group M22 is a sporadic group with order 443520 and it is a subgroup of the monstergroup. | The Mathieu group M22 is a sporadic group with order 443520 and it is a subgroup of the monstergroup. | ||
| Line 39: | Line 39: | ||
Relations; | Relations; | ||
| − | Gb:=NC.GB(Relations,31,1, | + | Gb:=NC.GB(Relations,31,1,100,1000); |
Gb; | Gb; | ||
| + | |||
====Example in Symbolic Data Format==== | ====Example in Symbolic Data Format==== | ||
<FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier"> | <FREEALGEBRA createdAt="2014-03-26" createdBy="strohmeier"> | ||
| Line 52: | Line 53: | ||
<ncpoly>(a*b*b)^5-1</ncpoly> | <ncpoly>(a*b*b)^5-1</ncpoly> | ||
<Comment>relation: (abb)^5=1</Comment> | <Comment>relation: (abb)^5=1</Comment> | ||
| − | <ncpoly>a*b*a*b*b*b*b*a*b*b*b | + | <ncpoly>(a*b*a*b*b*b*b*a*b*b*b)^3-1</ncpoly> |
<Comment>commutator: [a,bab]=ababb^3ab^3</Comment> | <Comment>commutator: [a,bab]=ababb^3ab^3</Comment> | ||
<Comment>relation [a,bab]^3=1</Comment> | <Comment>relation [a,bab]^3=1</Comment> | ||
| − | <ncpoly>a*b*a*b*a*b*b*b | + | <ncpoly>(a*b*a*b*a*b*b*b)^5-1</ncpoly> |
<Comment>relation (ababab^-1)^5=1</Comment> | <Comment>relation (ababab^-1)^5=1</Comment> | ||
</basis> | </basis> | ||
Latest revision as of 21:00, 22 April 2014
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>
