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Examples


Example GrpMatInf_IsFiniteMatrixGroupFQ (H61E1)

> Q := Rationals ();
> F<t>:= RationalFunctionField (Q);
> M:= MatrixAlgebra (F, 3);
> a:= M![-1, 2*t^2, -2*t^4 - 2*t^3 - 2*t^2, 0, 1, 0, 0, 0, 1];
> b:= M![1, 0, 0, 1/t^2, -1, (2*t^3 - 1)/(t - 1), 0, 0, 1];
> c:= M![t, -t^3 + t^2, t^5 - t^2 - t, t^2, -t^4, (t^8 - t^5 + 1)/
> (t^2 - t), (t - 1)/t, -t^2 + t, t^4 - t];
> G:= sub<GL(3,F)|a,b,c>;
> IsFinite(G);
true
> flag, H := IsomorphicCopy(G);
> H;
MatrixGroup(3, GF(3)) 
Generators:
    [2 2 1]
    [0 1 0]
    [0 0 1]


    [1 0 0]
    [1 2 0]
    [0 0 1]


    [2 2 2]
    [1 2 0]
    [2 1 2]
> #H;
48

Example GrpMatInf_IsFiniteMatrixGroupFF (H61E2)

> F<t>:= RationalFunctionField (GF(5));
> M:= MatrixAlgebra (F, 6);
> a:= M![2, 2*t^2, 4, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0,
> 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1];
> b:= M![(4*t + 4)/t, 4*t, (t + 1)/t, 0, t, t^2 + t, 0, 4, 0, 0, 0,
> 1/t, 4/t, t^2 + 4*t, 1/t, 0, 0, 0, 0, 4*t, 0, 0, 0, 0, 0, 0, 4, 4,
> 0, 0, 0, 0, 0, 4, 0, 0];
> G:= sub<GL(6,F)|a,b>;
> IsFinite(G);
true
> flag, H := IsomorphicCopy (G);
> flag;
true
> H;
MatrixGroup(6, GF(5)) of order 2^7 * 3 * 5^4 * 31 
Generators:
    [2 2 4 1 0 0]
    [0 2 0 0 0 0]
    [0 0 1 1 0 0]
    [0 0 0 1 0 0]
    [0 0 0 0 1 1]
    [0 0 0 0 0 1]


    [3 4 2 0 1 2]
    [0 4 0 0 0 1]
    [4 0 1 0 0 0]
    [0 4 0 0 0 0]
    [0 0 4 4 0 0]
    [0 0 0 4 0 0]
> #H;
7440000

Example GrpMatInf_IsFiniteMatrixGroupFF (H61E3)

>  L<t> := RationalFunctionField (GF (5^2));
>  G := GL (2, L);
>  a := G![t,1,0,-1];
>  b:= G![t/(t + 1), 1, 0, 1/t];
>  H := sub <GL(2, L) | a, b>;
> f :=IsFinite(H);
> f;
false
> IsSolubleByFinite (H);
true
> IsCompletelyReducible (H);
false

Example GrpMatInf_IsFiniteMatrixGroupFF (H61E4)

> G := MatrixGroup<3, IntegerRing() |
>  [ 5608, 711, -711, 6048, 766, -765, 1071, 135, -134 ],
>  [ 1, -2415, 5475, 0, 4471, -10140, 0, 780, -1769 ],
>  [ 5743, -5742, 639, -576, 577, -72, -711, 711, -80 ],
>  [ 526168, -618507, 729315, 621984, -731138, 862125,
>        274455, -322620, 380419      ]  ,
>  [ 648226, -4621455, 9226791, 660687, -4710305, 9404184,
>        85626, -610473, 1218820 ],
>  [ 32581, -39465, 46350, 53100, -64319, 75540, 24210,
>        -29325, 34441 ]>;
> IsFinite (G);
false
> IsSolubleByFinite (G);
false
> IsNilpotentByFinite (G);
false
> time IsCentralByFinite (G);
false
> IsAbelianByFinite (G);
false


Example GrpMatInf_IsFiniteMatrixGroupFF (H61E5)

> Q<z> := QuadraticField(5);
> O<w> := sub< MaximalOrder(Q) | 7 >;
> G := GL(2, Q);
> x := G![1,1+w,0,w];
> y :=  G![-1/2, 2, 2 + w, 5 + w^2];
> H:=sub<G | x, y>;
> IsFinite (H);
false
> IsSolubleByFinite (H);
false

Example GrpMatInf_IsFiniteMatrixGroup (H61E6)

>  R<x> := PolynomialRing(Integers());
>  K<y> := NumberField(x^4-420*x^2+40000);
>  G := GL (2, K);
>  a := G![y,1,0,-1];
>  b:= G![y/(y + 1), 1, 0, 1/y];
>  H := sub <GL(2, K) | a, b>;
> time IsFinite(H);
false

Example GrpMatInf_IsFiniteMatrixGroupF (H61E7)


> /* example over algebraic extension of a function field */
>                                                           
> R<u> := FunctionField (Rationals ());                     
> v := u; w := -2 * v;                                      
> Px<X> := PolynomialRing (R);                              
> Py<Y> := PolynomialRing (R);                              
> f :=  Y^2- 3 * u * X * Y^2  + v * X^3;                    
> facs := Factorisation (f);                                
> F:=ext <R | facs[2][1]>;                                  
> F;                                                        
Algebraic function field defined over Univariate rational function 
field over Rational Field by Y - 1/2/u
>                                                                                            
> n := 3;                                                                                    
> G:= GL(n,F);                                                                               
> Z := 4 * X * Y;                                                                            
> MA:= MatrixAlgebra(F,n);                                                                   
> h1:= Id(MA);                                                                               
> h1[n][n]:= (X^2+Y+Z+1);                                                                    
> h1[1][n]:= X+1;                                                                            
> h1[1][n]:= X+1;                                                                            
> h1[1][1]:=(Z^5-X^2*Z+Z*X*Y);                                                               
> h1[2][1]:=1-X*Y*Z;                                                                         
> h1[2][n]:= X^20+X*Y^15+Y^10+Z^4*Y*X^5+1;                                                   
> h2:= Id(MA);                                                                               
> h2[n][n]:= (X^7+Z^6+1);                                                                    
> h2[1][n]:= X^2+X+1;                                                                        
> h2[1][1]:=(Y^3+X^2+X+1);                                                                   
> h2[1][1]:=(Y^3+X^2+X+1);                                                                   
> h2[2][1]:=1-X^2;                                                                           
> h2[2][n]:= X^50+Y^35+X^20+X^13+Y^2+1;                                                      
> G := sub< GL(n, F) | h1, h2>;                                                              
> G;            
MatrixGroup(3, F)
Generators:
    [1/u^10 0 (u + 1/2)/u]
    [(u^4 - 1/4)/u^4 1 (u^20 + 1/1024*u^10 + 1/64*u^6 + 1/65536*u^4 +
        1/1048576)/u^20]
    [0 0 (u^2 + 1/2*u + 5/4)/u^2]


    [(u^3 + 1/2*u^2 + 1/4*u + 1/8)/u^3 0 (u^2 + 1/2*u + 1/4)/u^2]
    [(u^2 - 1/4)/u^2 1 (u^50 + 1/4*u^48 + 1/8192*u^37 + 1/1048576*u^30 +
        1/34359738368*u^15 + 1/1125899906842624)/u^50]
    [0 0 (u^12 + 1/128*u^5 + 1)/u^12]
> time IsFinite(G);
false
Time: 0.010
> time IsSolubleByFinite (G);
true

Example GrpMatInf_IsFiniteMatrixGroupF (H61E8)

> F := GF(2);                             
> P := PolynomialRing (F);                
> P<t> := PolynomialRing (F);             
> F := ext < F | t^2+t+1>;                
> G := GL (2, FunctionField (F));         
> a := G![1,1/t, 0, 1];                   
> b := [1,1/(t + 1), 0, 1];               
> c := [1,1/(t^2 + t + 1), 0, 1];         
> d := [1,1/(t^2 + t), 0, 1];             
> G := sub < G | a,b,c,d>;                
> time IsFinite (G);
true
> f, I, tau := IsomorphicCopy (G);
> f;
true

Example GrpMatInf_IsNilpotentMatrixGroupF (H61E9)


> // irreducible but (evidently) imprimitive
> K<w> := QuadraticField (2);
> G := MatrixGroup< 8, K |                  
>     [1/2*w,1/2*w,0,0,0,0,0,0,-1/2*w,1/2*w,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,
>      0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,
>      0,0,0,1],                                                            
>     [1,0,0,0,0,0,0,0,0,-1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,
>      0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1],        
>     [0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,0,0,0,0, 
>     0,0,0,0,1,0,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0] >;
> G;                                                                        
MatrixGroup(8, K)                                                           
Generators:                                                                 
    [ 1/2*w  1/2*w        0        0        0        0        0        0]   
    [-1/2*w  1/2*w        0        0        0        0        0        0]   
    [       0        0        1        0        0        0        0        0]
    [       0        0        0        1        0        0        0        0]
    [       0        0        0        0        1        0        0        0]
    [       0        0        0        0        0        1        0        0]
    [       0        0        0        0        0        0        1        0]
    [       0        0        0        0        0        0        0        1]


    [ 1  0  0  0  0  0  0  0]
    [ 0 -1  0  0  0  0  0  0]
    [ 0  0  1  0  0  0  0  0]
    [ 0  0  0  1  0  0  0  0]
    [ 0  0  0  0  1  0  0  0]
    [ 0  0  0  0  0  1  0  0]
    [ 0  0  0  0  0  0  1  0]
    [ 0  0  0  0  0  0  0  1]


    [0 0 1 0 0 0 0 0]
    [0 0 0 1 0 0 0 0]
    [0 0 0 0 1 0 0 0]
    [0 0 0 0 0 1 0 0]
    [0 0 0 0 0 0 1 0]
    [0 0 0 0 0 0 0 1]
    [1 0 0 0 0 0 0 0]
    [0 1 0 0 0 0 0 0]
> IsIrreducibleFiniteNilpotent(G);
true
> r, B := IsPrimitiveFiniteNilpotent(G);
> r;
false
> #B;
2

Example GrpMatInf_IsNilpotentMatrixGroupF (H61E10)

> M:= MatrixAlgebra (GF(17), 4);
> a:= M![5, 5, 3, 3, 0, 5, 0, 3, 16, 16, 14, 14, 0, 16, 0, 14];
> b:= M![9, 9, 0, 0, 0, 9, 0, 0, 10, 10, 8, 8, 0, 10, 0, 8];
> G:= sub<GL(4,17)|a,b>;
> IsNilpotent(G);
true
> SylowSystem (G);
[
    MatrixGroup(4, GF(17))
    Generators:
        [ 5  0  3  0]
        [ 0  5  0  3]
        [16  0 14  0]
        [ 0 16  0 14]


        [ 9  0  0  0]
        [ 0  9  0  0]
        [10  0  8  0]
        [ 0 10  0  8],


    MatrixGroup(4, GF(17))
    Generators:
        [ 1  1  0  0]
        [ 0  1  0  0]
        [ 0  0  1  1]
        [ 0  0  0  1]
]
> Order(G);
8704

Example GrpMatInf_IsNilpotentMatrixGroupF (H61E11)

> R<s>:= QuadraticField(-1);
> F<t>:= FunctionField(R);
> M:= MatrixAlgebra (F, 2);
> a:= M![-s*t^2 + 1, s*t^3, -s*t, s*t^2 + 1];
> b:= M![t^2 - 3*t + 1, 0, 0, t^2 - 3*t + 1];
> G:= sub<GL(2,F)|a,b>;
> IsNilpotent(G);
true
> IsFinite(G);
false
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Version: V2.19 of Wed Apr 24 15:09:57 EST 2013