# Difference between revisions of "ApCoCoA-1:Higman groups"

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− | === <div id="Higman_groups">[[:ApCoCoA:Symbolic data#Higman_group|Higman | + | === <div id="Higman_groups">[[:ApCoCoA:Symbolic data#Higman_group|Higman Group]]</div> === |

==== Description ==== | ==== Description ==== | ||

The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the | The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the | ||

Line 68: | Line 68: | ||

<ncpoly>h*a*d-a*a</ncpoly> | <ncpoly>h*a*d-a*a</ncpoly> | ||

</basis> | </basis> | ||

− | <Comment>The LLex has 144 elements</Comment> | + | <Comment>The partial LLex Gb has 144 elements</Comment> |

<Comment>Higman_group</Comment> | <Comment>Higman_group</Comment> | ||

</FREEALGEBRA> | </FREEALGEBRA> |

## Latest revision as of 20:50, 22 April 2014

#### Description

The Higman group is an infinite finitely presented group with no non-trivial finite quotients and is generated by the elements a,b,c and d with the following relations:

H = <a,b,c,d | a^{-1}ba = b^{2}, b^{-1}cb = c^{2}, c^{-1}dc = d^{2}, d^{-1}ad = a^{2}>

#### Reference

Graham Higman, "A finitely generated infinite simple group", Journal of the London Mathematical Society. Second Series 26 (1): 61–64, 1951.

#### Computation

/*Use the ApCoCoA package ncpoly.*/ // a is inverse to e, b is inverse to f, c is inverse to g and d is inverse to h Use ZZ/(2)[a,b,c,d,e,f,g,h]; NC.SetOrdering("LLEX"); Define CreateRelationsHigman() Relations:=[]; // add the relations of the inverse elements Append(Relations,[[a,e],[1]]); Append(Relations,[[e,a],[1]]); Append(Relations,[[b,f],[1]]); Append(Relations,[[f,b],[1]]); Append(Relations,[[c,g],[1]]); Append(Relations,[[g,c],[1]]); Append(Relations,[[d,h],[1]]); Append(Relations,[[h,d],[1]]); // add the relation a^{-1}ba = b^2 Append(Relations,[[e,b,a],[b^2]]); // add the relation b^{-1}cb = c^2 Append(Relations,[[f,c,b],[c^2]]); // add the relation c^{-1}dc = d^2 Append(Relations, [[g,d,c],[d^2]]); // add the relation d^{-1}ad = a^2 Append(Relations, [[h,a,d],[a^2]]); Return Relations; EndDefine; Relations:=CreateRelationsHigman(); Relations; Gb:=NC.GB(Relations,31,1,100,1000); Gb; Size(Gb);

#### Example in Symbolic Data Format

<FREEALGEBRA createdAt="2014-01-20" createdBy="strohmeier"> <vars>a,b,c,d,e,f,g,h</vars> <uptoDeg>5</uptoDeg> <basis> <ncpoly>a*e-1</ncpoly> <ncpoly>e*a-1</ncpoly> <ncpoly>b*f-1</ncpoly> <ncpoly>f*b-1</ncpoly> <ncpoly>c*g-1</ncpoly> <ncpoly>g*c-1</ncpoly> <ncpoly>d*h-1</ncpoly> <ncpoly>h*d-1</ncpoly> <ncpoly>e*b*a-b*b</ncpoly> <ncpoly>f*c*b-c*c</ncpoly> <ncpoly>g*d*c-d*d</ncpoly> <ncpoly>h*a*d-a*a</ncpoly> </basis> <Comment>The partial LLex Gb has 144 elements</Comment> <Comment>Higman_group</Comment> </FREEALGEBRA>