> 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
> 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
> 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
> 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
> 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
> 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 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
> 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
> // 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
> 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
> 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[Next][Prev] [Right] [Left] [Up] [Index] [Root]