##### # # This is GAP code # # The file is ~steger/donald/SU21/C2p2/C2p2-3-proj.gap # # See file ~steger/donald/SU21/C2p2/D.txt for the definition of the # division algebra $\D$ and for all the notation. # ##### Read("/home/steger/donald/SU21/C2p2/C2p2-proj.gap"); Read("/home/steger/donald/SU21/NormElem.gap"); ##### # # Free group # ###### #From: "Donald Cartwright" #To: "steger@uniss.it" #Subject: and results for (C2,p=2,\emptyset) #Date: Mon, 7 Apr 2008 21:36:13 +1000 FreeGp:=FreeGroup("Z","A","B"); Zf:=FreeGp.1; Af:=FreeGp.2; Bf:=FreeGp.3; RelnList:=[ Zf^3, (Bf*Zf)^3, Af*Zf*Bf*Zf^-1*Af^-1*Zf^-1*Af^-1*Zf*Bf^-1, Af^-1*Bf^-1*Zf^-1*Af^-1*Zf*Bf^-1*Zf*Af^-1*Bf^-1*Zf^-1, Bf^-1*Zf*Af^-1*Bf^-1*Zf*Bf*Zf^-1*Af*Bf^-1*Zf^-1*Bf^-1, Bf*Zf^-1*Af*Zf^-1*Bf^-1*Af*Bf^-1*Zf*Af*Zf*Bf, Af^-2*Zf^-1*Af^-1*Zf^-1*Bf^-1*Zf*Af^-2*Bf*Zf, (Af^-1*Zf^-1*Af^-1*Bf)^3, Bf^-1*Zf*Bf^-1*Af*Zf^-1*Af^-2*Bf*Zf^-1*Af^-1*Bf*Zf*Af^-1, Zf*Af^-1*Bf*Zf*Bf^-1*Zf*Af^-1*Zf*Bf^2*Af*Zf^-1*Bf, Zf^-1*Af^-1*Zf*Af^-1*Zf^-1*Af^-1*Bf*Af*Zf^-1*Bf^-1*Af*Zf^-1*Af^-2, Af^-1*Bf^-1*Zf^-1*Af^-1*Zf^-1*Af*Zf*Af^2*Zf*Af*Zf*Af*Zf^-1*Bf, Af^-1*Bf*Zf^-1*Bf*Af*Bf^-1*Zf*Af^-1*Bf^-2*Zf*Af^-2*Bf*Zf^-1, Bf^-1*Af*Bf^-1*Zf*Bf^-1*Af*Zf^-1*Bf^-1*Zf^-2*Af^-1*Bf^-2*Zf*Af^-1*Zf, Af*Zf^-1*Bf^2*Zf^-1*Af^-2*Bf*Zf*Af^-1*Zf*Bf*Af*Zf*Bf^-1*Zf^-1, Bf^-2*Af^2*Zf^-1*Bf^2*Af*Zf*Bf^-1*Zf^-1*Af*Bf^-1*Zf*Af^-1*Bf^-1, Af^-1*Bf*Zf^-1*Bf*Af*Zf*Af^-1*Bf*Zf*Af^-1*Bf^-1*Zf^-1*Af^-1*Zf*Bf*Af*Zf^-1*Bf*Af^-1*Bf^-1, Zf*Af^-1*Bf*Zf^-1*Af*Bf*Zf*Bf*Af^-1*Bf*Zf*Bf*Af*Zf*Bf^-1*Zf^-1*Af*Bf*Zf*Bf*Af^-1 ]; GammaBarFP:=FreeGp/RelnList;; ZFP:=GammaBarFP.1; AFP:=GammaBarFP.2; BFP:=GammaBarFP.3; # Important subgroups of GammaBarFP index3aFP:=Group(ZFP, AFP); index3bFP:=Group(AFP, BFP*ZFP, ZFP*BFP); index3cFP:=Group(AFP, ZFP*BFP^-1, ZFP^-1*BFP); index3dFP:=Group(AFP, BFP, ZFP*AFP*ZFP^-1); index9FP:=Group(AFP, ZFP*AFP*ZFP^-1, ZFP*BFP*ZFP^-1*BFP^-1, BFP*AFP*BFP^-1); # Check the indexes of the subgroups Index(GammaBarFP,index3aFP) = 3; Index(GammaBarFP,index3bFP) = 3; Index(GammaBarFP,index3cFP) = 3; Index(GammaBarFP,index3dFP) = 3; Index(GammaBarFP,index9FP) = 9; # Check normality IsNormal(GammaBarFP,index3aFP); IsNormal(GammaBarFP,index3bFP); IsNormal(GammaBarFP,index3cFP); IsNormal(GammaBarFP,index3dFP); IsNormal(GammaBarFP,index9FP); # Free group versions of these subgroups FPtoFGSubgp:=g->Group(List(GeneratorsOfGroup(g),UnderlyingElement)); index3aFG:=FPtoFGSubgp(index3aFP); index3bFG:=FPtoFGSubgp(index3bFP); index3cFG:=FPtoFGSubgp(index3cFP); index3dFG:=FPtoFGSubgp(index3dFP); index9FG:=FPtoFGSubgp(index9FP); ##### # # Matrix generators of $(C2,p=2,\{3\})$ # ##### #From: "Donald Cartwright" #To: "steger@uniss.it" #Subject: Generators for one of the (C_2,p=2) groups? #Date: Thu, 27 Mar 2008 09:39:47 +1100 ADM:= [[-1/3*G*Zm^2+1/3*(-G-1)*Zm+1/6*G*Sqrt(-3)+1/6*(G+2), (1/6*(-2*G-3)*Sqrt(-3)+1/6*(-4*G+1))*Zm^2+(1/3*(G+1)*Sqrt(-3)+1/3*(G+1))*Zm+1/6*(-2*G+1)*Sqrt(-3)+1/6*(-4*G-3), (1/6*(-2*G-1)*Sqrt(-3)-1/6)*Zm^2+(1/6*(G+2)*Sqrt(-3)+1/2*G)*Zm+-1/6*G*Sqrt(-3)+1/6*(-3*G-2)], [(1/6*(-G-1)*Sqrt(-3)+1/6*(-3*G-1))*Zm^2+(1/6*(-2*G-1)*Sqrt(-3)-1/2)*Zm+1/6*(-2*G-1)*Sqrt(-3)+1/6, (1/6*G*Sqrt(-3)+1/6*G)*Zm^2+(1/6*(-G-1)*Sqrt(-3)+1/6*(G+1))*Zm+1/6*G*Sqrt(-3)+1/6*(G+2), (1/6*(3*G+1)*Sqrt(-3)+1/6*(-G-5))*Zm^2+1/3*(-2*G-2)*Zm+1/6*(-2*G+1)*Sqrt(-3)+1/6*(-4*G-3)], [(1/6*(3*G+2)*Sqrt(-3)+1/6*(-G+4))*Zm^2+(1/6*G*Sqrt(-3)+1/6*(5*G+2))*Zm+1/6*(-3*G-1)*Sqrt(-3)+1/6*(-G+3), (1/6*(2*G+1)*Sqrt(-3)-1/6)*Zm^2+(1/6*(G-1)*Sqrt(-3)+1/2*(G+1))*Zm+1/6*(-2*G-1)*Sqrt(-3)+1/6, (-1/6*G*Sqrt(-3)+1/6*G)*Zm^2+(1/6*(G+1)*Sqrt(-3)+1/6*(G+1))*Zm+1/6*G*Sqrt(-3)+1/6*(G+2)]]; BDM:=[ [(-1/6*G*Sqrt(-3)+1/6*(-G-2))*Zm+1/6*Sqrt(-3)+1/6*(-2*G-1), (-1/6*Sqrt(-3)+1/6*(2*G+3))*Zm^2+(1/6*Sqrt(-3)+1/6*(-2*G+1))*Zm+-1/3*G*Sqrt(-3)+1/3, (1/6*(G+1)*Sqrt(-3)+1/6*(G+3))*Zm^2+(-1/3*Sqrt(-3)-1/3)*Zm+1/6*(G+1)*Sqrt(-3)+1/6*(5*G-1)], [2/3*G*Zm^2+(1/6*(2*G-1)*Sqrt(-3)+1/6)*Zm+1/6*(G+3)*Sqrt(-3)+1/6*(-G-1), (-1/6*Sqrt(-3)+1/6*(2*G+1))*Zm+1/6*Sqrt(-3)+1/6*(-2*G-1), (1/6*(-G-1)*Sqrt(-3)+1/6*(-G-3))*Zm^2+(-1/6*G*Sqrt(-3)+1/6*(G-2))*Zm+-1/3*G*Sqrt(-3)+1/3], [-2/3*G*Zm^2+(1/6*G*Sqrt(-3)+1/6*(G-2))*Zm+1/12*(G-1)*Sqrt(-3)+1/12*(3*G+7), (-1/3*G*Sqrt(-3)-1/3*G)*Zm^2+(1/6*(-G+1)*Sqrt(-3)+1/6*(-3*G+1))*Zm+1/6*(G+3)*Sqrt(-3)+1/6*(-G-1), (1/6*(G+1)*Sqrt(-3)+1/6*(-G+1))*Zm+1/6*Sqrt(-3)+1/6*(-2*G-1)]]; ZDM:=[[Zm, 0, 0], [0,Zm^4, 0], [0, 0,Zm^7]]; ZDM in DM; ADM in DM; BDM in DM; IsOne(iotaDM(ZDM)*ZDM); IsOne(iotaDM(ADM)*ADM); IsOne(iotaDM(BDM)*BDM); GammaBarDM:=Group(ZDM,ADM,BDM);; DMHom:=GroupHomomorphismByImages(FreeGp,GammaBarDM, [Zf,Af,Bf],[ZDM,ADM,BDM]);; ForAll(RelnList,x->IsPOne(x^DMHom)); ##### # # Construct matrices for the group generators acting on~$\D$ by # conjugation # ###### DMtoPDM:=mtx->TransposedMat( List(basisDM,bmtx-> Coefficients(basisDM,mtx*bmtx*mtx^-1) )); ZPDM:=DMtoPDM(ZDM);; APDM:=DMtoPDM(ADM);; BPDM:=DMtoPDM(BDM);; GammaBarPDM:=Group(ZPDM,APDM,BPDM);; FGtoPDM:=GroupHomomorphismByImages(FreeGp,GammaBarPDM, [Zf,Af,Bf],[ZPDM,APDM,BPDM]);; ForAll(RelnList,x->IsOne(x^FGtoPDM)); ###### # # Find the matrices for the group generators acting by conjugation on # the self-adjoint, traceless part of~$\D$. # ##### PDMtoPSA:=mtx->TransposedMat( List(DMSelfAdjBasis,SAVec-> Coefficients(DMSelfAdjBasis,mtx*SAVec) )); ZPSA0:=PDMtoPSA(ZPDM);; APSA0:=PDMtoPSA(APDM);; BPSA0:=PDMtoPSA(BPDM);; ##### # # Adjust the basis of the self-adjoint part of~$\D$ so that # the group generators correspond to integral matrices. # ##### GammaBarPSA0:=Group(ZPSA0,APSA0,BPSA0);; ILatPSA0:=InvariantLattice(GammaBarPSA0)^-1; ZPSA:=ZPSA0^ILatPSA0; APSA:=APSA0^ILatPSA0; BPSA:=BPSA0^ILatPSA0; GammaBarPSA:=Group(ZPSA,APSA,BPSA);; FGtoPSA:=GroupHomomorphismByImages(FreeGp,GammaBarPSA, [Zf,Af,Bf],[ZPSA,APSA,BPSA]);; ForAll(RelnList,x->IsOne(x^FGtoPSA)); #APSA:=[ #[67, -60,-118, 430, -29,-175, 65, 245,-191,-1090,3820,-1762,1174,1575,4020, 9410], #[50, -70, -35, -71, 7, 49, 48, 96, 48, -236,1089, -318, 224, 722, 976, 1630], #[12, -3, -6, -67, -21, 26, 54, 67, 95, -330,1270, -93, 163, 150, 100, 1285], #[61, -81, -65, 94, 5, -18, 41, 137, -42, -501,1978, -881, 573,1210,2304, 4510], #[11, -22, 8,-133, 10, 65, 11, -2, 74, 67, -83, 184,-104, 76,-296, -897], #[78, -92,-103, 267, -1, -94, 56, 213,-114, -869,3185,-1497, 975,1695,3645, 7810], #[ 8, -17, -9, 26, 11, -5, -8, 6, -30, 11, -34, -114, 45, 177, 347, 353], #[-7, 8, 21,-118, 19, 41, 18, -2, 115, 6,-132, 269,-135,-217,-802, -933], #[25, -34, -22, 3, -2, 9, 20, 50, 2, -178, 766, -283, 193, 466, 801, 1485], #[-5, 5, 3, 17, 16, -14, 2, 8, 22, -80, 118, -55, 43, -2, -15, 514], #[-1, 1, 2, -8, 1, 3, 0, -2, 8, -3, 8, 14, -5, -7, -39, -25], #[ 1, -6, 4, -32, 20, 12, 0, 0, 28, 41,-224, 70, -56, -26,-211, -377], #[15, -30, -3, -75, 5, 44, 10, 5, 10, 140,-360, 135,-112, 60,-130,-1070], #[ 3, -7, 0, -18, 2, 11, -1, -3, 0, 46,-125, 36, -32, 17, -28, -296], #[ 0, 0, 3, -24, 1, 10, 2, -2, 17, 10, -21, 48, -25, -21,-112, -199], #[-3, 5, 0, 21, 3, -12, -2, 1, -4, -25, 28, -33, 21, -17, 21, 220]]; #BPSA:=[ #[26, -30, -11, -83, -4, 28, 64, 93, 156, -430, 1530,-201, 240, 390, 380, 2025], #[28, -13, -44, 116, 48, -92, 123, 248, 183,-1027, 2840,-792, 638, 535, 859, 5861], #[-5, -1, -3, 77, -19, -16, -62, -66,-124, 193, -425,-115, 14, 175, 630, -70], #[19, 1, -27, 41, -7, -47, 101, 150, 99, -509, 1477,-254, 246, 22, 53, 2178], #[ 3, 9, -19, 104, 12, -60, 27, 71, -5, -270, 661,-293, 194, 83, 319, 1792], #[34, -20, -32, -9, 8, -28, 126, 194, 184, -714, 2160,-378, 370, 295, 295, 3295], #[-4, 17, 2, -17, -20, -2, 29, 13, 9, 39, -147, 189,-111,-331, -566, -977], #[ 3, 0, 1, -35, 14, 7, 19, 25, 47, -20, -53, 68, -47, -53, -315, -248], #[ 7, 0, -10, 16, -5, -17, 37, 53, 30, -167, 486, -79, 78, -10, 4, 673], #[-9, 17, 22,-127, -36, 57, 4, -66, -9, 457,-1331, 632,-440,-606,-1263,-3884], #[-2, 3, 2, -3, -6, 3, -6, -15, -18, 79, -226, 65, -54, -69, -97, -495], #[ 4, 5, -2, -34, 11, -2, 49, 62, 85, -168, 369, 42, 7,-121, -431, 236], #[ 4, 5, -11, 30, 10, -32, 60, 95, 80, -370, 975,-180, 178, -5, -30, 1670], #[ 0, 3, -3, 16, 0, -11, 11, 18, 7, -70, 181, -42, 37, -17, 5, 336], #[ 2, -2, -4, 18, 8, -10, 3, 16, 7, -77, 201, -93, 63, 83, 148, 586], #[-1, 1, 5, -32, -4, 15, -1, -14, 5, 84, -239, 117, -80, -85, -220, -697]]; #ZPSA:=[ #[ 1, 0, -15, 101, 1, -38, -9, 14, -68,-128, 425,-294, 184, 190, 660, 1475], #[ 0, 1, -3, 15, 3, -6, -13, 4, 0, -60, 233, -78, 61, 83, 162, 510], #[ 0, 0, -23, 186, -30, -62, -52, -36,-189, 14, 180,-404, 215, 300,1265, 1490], #[ 0, 0, 0, -11, 0, 14, -37, -41, -45, 163,-373, 68, -70, 3, 28, -716], #[ 0, 0, 0, 0, -14, 8, -31, -38, -45, 136,-262, 50, -49, 12, 105, -566], #[ 0, 0, 0, -15, 15, 10, -15, -15, -15, 90,-285, 60, -58, -45,-125, -505], #[ 0, 0, 3, -26, -7, 16, -21, -30, -17, 134,-312, 108, -83, -40,-102, -792], #[ 0, 0, -6, 50, -5, -20, -11, -12, -57, 80,-266, -26, -13, -45, 134, -197], #[ 0, 0, 3, -30, 0, 17, -12, -19, -4, 85,-197, 82, -60, -23,-108, -549], #[ 0, 0, 0, -1, -11, 7, -23, -43, -63, 243,-683, 148,-139,-139,-129,-1342], #[ 0, 0, 0, 0, -3, 2, -6, -10, -14, 46,-115, 22, -22, -13, 3, -224], #[ 0, 0, 0, 0, 0, -1, -6, -7, -8, 61,-203, 53, -45, -65,-101, -405], #[ 0, 0, 0, 0, 0, -3, 0, 15, 30,-105, 345, -60, 64, 75, 75, 585], #[ 0, 0, 0, 0, 0, 0, -3, 0, 3, -12, 51, -12, 11, 22, 29, 96], #[ 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0], #[ 0, 0, 0, 0, 0, 0, 0, -3, -6, 30,-105, 24, -22, -33, -46, -197]]; ###### # # Do calculations $\mod 2$ # ###### toMod2:=x->x*One(GF(2)); toMod2M:=mtx->fOnM(toMod2,mtx); ZPSAMod2:=toMod2M(ZPSA);; APSAMod2:=toMod2M(APSA);; BPSAMod2:=toMod2M(BPSA);; GammaBarPSAMod2:=Group(ZPSAMod2,APSAMod2,BPSAMod2);; FGtoPSAMod2:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod2, [Zf,Af,Bf],[ZPSAMod2,APSAMod2,BPSAMod2]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod2)); Read("/home/steger/donald/SU21/NormElem2.gap"); Mod2H1:=Group(NormElem2(GammaBarPSAMod2,2)); Size(Mod2H1)=64; StructureDescription(Mod2H1)="C2 x C2 x C2 x C2 x C2 x C2"; Mod2Q1:=Group(Mod2H1LAL); not Size(Mod2Q1)*Size(Mod2H1) = Size(GammaBarPSAMod2); Mod2NHNS1:=NaturalHomomorphismByNormalSubgroup(GammaBarPSAMod2,Mod2H1);; Mod2Q1:=Image(Mod2NHNS1); Size(Mod2Q1)*Size(Mod2H1) = Size(GammaBarPSAMod2); Mod2H2:=Group(NormElem2(Mod2QQ1,2)); Size(Mod2H2)=64; StructureDescription(Mod2H2)="C2 x C2 x C2 x C2 x C2 x C2"; Mod2ASA2:= AbelianSubfactorAction(Mod2Q1,Mod2H2,Group(One(Mod2H2))); Mod2Q2:=Image(Mod2ASA2[1]); Size(Mod2Q2)*Size(Mod2H2) = Size(Mod2QQ1); StructureDescription(Mod2Q2) = "C3 x (C7 : C3)"; PSAMod2Hom63:=CompositionMapping(Mod2ASA2[1],Mod2NHNS1); PSAMod2Q63:=Range(PSAMod2Hom63); StructureDescription(PSAMod2Q63) = "C3 x (C7 : C3)";; index3aPSAMod2:=Image(FGtoPSAMod2,index3aFG); index3bPSAMod2:=Image(FGtoPSAMod2,index3bFG); index3cPSAMod2:=Image(FGtoPSAMod2,index3cFG); index3dPSAMod2:=Image(FGtoPSAMod2,index3dFG); index9PSAMod2:=Image(FGtoPSAMod2,index9FG); Index(GammaBarPSAMod2,index3aPSAMod2) = 3; Index(GammaBarPSAMod2,index3bPSAMod2) = 3; Index(GammaBarPSAMod2,index3cPSAMod2) = 3; Index(GammaBarPSAMod2,index3dPSAMod2) = 3; Index(GammaBarPSAMod2,index9PSAMod2) = 9; Intersection(index3aPSAMod2,index3bPSAMod2) = index9PSAMod2; Intersection(index3aPSAMod2,index3cPSAMod2) = index9PSAMod2; Intersection(index3aPSAMod2,index3dPSAMod2) = index9PSAMod2; Intersection(index3bPSAMod2,index3cPSAMod2) = index9PSAMod2; Intersection(index3bPSAMod2,index3dPSAMod2) = index9PSAMod2; Intersection(index3cPSAMod2,index3dPSAMod2) = index9PSAMod2; index3aPSAMod2Q63:=Image(PSAMod2Hom63,index3aPSAMod2); index3bPSAMod2Q63:=Image(PSAMod2Hom63,index3bPSAMod2); index3cPSAMod2Q63:=Image(PSAMod2Hom63,index3cPSAMod2); index3dPSAMod2Q63:=Image(PSAMod2Hom63,index3dPSAMod2); index9PSAMod2Q63:=Image(PSAMod2Hom63,index9PSAMod2); Index(PSAMod2Q63,index3aPSAMod2Q63) = 3; Index(PSAMod2Q63,index3bPSAMod2Q63) = 3; Index(PSAMod2Q63,index3cPSAMod2Q63) = 3; Index(PSAMod2Q63,index3dPSAMod2Q63) = 3; Index(PSAMod2Q63,index9PSAMod2Q63) = 9; IsSubgroup(index3aPSAMod2Q63,Center(PSAMod2Q63)); not IsSubgroup(index3bPSAMod2Q63,Center(PSAMod2Q63)); not IsSubgroup(index3cPSAMod2Q63,Center(PSAMod2Q63)); not IsSubgroup(index3dPSAMod2Q63,Center(PSAMod2Q63)); ###### # # Do calculations $\mod 3$ # ###### toMod3:=x->x*One(GF(3)); toMod3M:=mtx->fOnM(toMod3,mtx); ZPSAMod3:=toMod3M(ZPSA);; APSAMod3:=toMod3M(APSA);; BPSAMod3:=toMod3M(BPSA);; GammaBarPSAMod3:=Group(ZPSAMod3,APSAMod3,BPSAMod3);; FGtoPSAMod3:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod3, [Zf,Af,Bf],[ZPSAMod3,APSAMod3,BPSAMod3]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod3)); #Size(GammaBarPSAMod3) = 8*9*10*9^5; time; # Mod3H1:=Group(NormElem2(GammaBarPSAMod3,3)); Size(Mod3H1)=9^3; StructureDescription(Mod3H1) = "C3 x C3 x C3"; Mod3Q1:=QGroup(GammaBarPSAMod3,Mod3H1,3); FGtoMod3Q1:=GroupHomomorphismByImages(FreeGp,Mod3Q1, [Zf,Af,Bf],GeneratorsOfGroup(Mod3Q1));; ForAll(RelnList,x->IsOne(x^FGtoMod3Q1)); Mod3H2:=ChSeriesMod3Q1[3]; Mod3H2LAL:=LinearActionLayer( Mod3Q1,GeneratorsOfGroup(Mod3Q1),Pcgs(Mod3H2)); Mod3Q2:=Group(Mod3H2LAL); FGtoMod3Q2:=GroupHomomorphismByImages(FreeGp,Mod3Q2, [Zf,Af,Bf],Mod3H2LAL);; ForAll(RelnList,x->IsOne(x^FGtoMod3Q2)); IdSmallGroup(Mod3Q2) = IdSmallGroup(SL(2,9)); AbelianInvariants(Mod3Q1) = []; index3aPSAMod3:=Image(FGtoPSAMod3,index3aFG); index3bPSAMod3:=Image(FGtoPSAMod3,index3bFG); index3cPSAMod3:=Image(FGtoPSAMod3,index3cFG); index3dPSAMod3:=Image(FGtoPSAMod3,index3dFG); index9PSAMod3:=Image(FGtoPSAMod3,index9FG); index3aPSAMod3 = GammaBarPSAMod3; index3bPSAMod3 = GammaBarPSAMod3; index3cPSAMod3 = GammaBarPSAMod3; index3dPSAMod3 = GammaBarPSAMod3; index9PSAMod3 = GammaBarPSAMod3; Intersection(index3aPSAMod3,index3bPSAMod3) = index9PSAMod3; Intersection(index3aPSAMod3,index3cPSAMod3) = index9PSAMod3; Intersection(index3aPSAMod3,index3dPSAMod3) = index9PSAMod3; Intersection(index3bPSAMod3,index3cPSAMod3) = index9PSAMod3; Intersection(index3bPSAMod3,index3dPSAMod3) = index9PSAMod3; Intersection(index3cPSAMod3,index3dPSAMod3) = index9PSAMod3; index3aMod3Q1:=Image(FGtoMod3Q1,index3aFG); index3bMod3Q1:=Image(FGtoMod3Q1,index3bFG); index3cMod3Q1:=Image(FGtoMod3Q1,index3cFG); index3dMod3Q1:=Image(FGtoMod3Q1,index3dFG); index9Mod3Q1:=Image(FGtoMod3Q1,index9FG); index3aMod3Q1 = Mod3Q1; index3bMod3Q1 = Mod3Q1; index3cMod3Q1 = Mod3Q1; index3dMod3Q1 = Mod3Q1; index9Mod3Q1 = Mod3Q1; index3aMod3Q2:=Image(FGtoMod3Q2,index3aFG); index3bMod3Q2:=Image(FGtoMod3Q2,index3bFG); index3cMod3Q2:=Image(FGtoMod3Q2,index3cFG); index3dMod3Q2:=Image(FGtoMod3Q2,index3dFG); index9Mod3Q2:=Image(FGtoMod3Q2,index9FG); index3aMod3Q2 = Mod3Q2; index3bMod3Q2 = Mod3Q2; index3cMod3Q2 = Mod3Q2; index3dMod3Q2 = Mod3Q2; index9Mod3Q2 = Mod3Q2; ###### # # Do calculations $\mod 5$ # ###### toMod5:=x->x*One(GF(5)); toMod5M:=mtx->fOnM(toMod5,mtx); ZPSAMod5:=toMod5M(ZPSA);; APSAMod5:=toMod5M(APSA);; BPSAMod5:=toMod5M(BPSA);; GammaBarPSAMod5:=Group(ZPSAMod5,APSAMod5,BPSAMod5);; FGtoPSAMod5:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod5, [Zf,Af,Bf],[ZPSAMod5,APSAMod5,BPSAMod5]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod5)); Mod5H1:=Group(NormElem2(GammaBarPSAMod5,5)); Size(Mod5H1)=5^8; time; #150437 StructureDescription(Mod5H1) = "C5 x C5 x C5 x C5 x C5 x C5 x C5 x C5"; Mod5Q1:=QGroup(GammaBarPSAMod5,Mod5H1,5); IsomorphismGroups(PGU(3,5),Mod5Q1); time; #28630 MTXMod5H1:=GModuleByMats(GeneratorsOfGroup(Mod5Q1),GF(5)); MTX.IsAbsolutelyIrreducible(MTXMod5H1); # Structure of GammaBarPSAMod5 seems to be # # 1 -> C5^8 -> GammaBarPSAMod5 -> PGU(3,5) -> 1 # # with a faithful action on C5^8. FGtoMod5Q1:=GroupHomomorphismByImages(FreeGp,Mod5Q1, [Zf,Af,Bf],GeneratorsOfGroup(Mod5Q1)); ForAll(RelnList,x->IsOne(x^FGtoMod5Q1)); index3aMod5Q1:=Image(FGtoMod5Q1,index3aFG); index3bMod5Q1:=Image(FGtoMod5Q1,index3bFG); index3cMod5Q1:=Image(FGtoMod5Q1,index3cFG); index3dMod5Q1:=Image(FGtoMod5Q1,index3dFG); index9Mod5Q1:=Image(FGtoMod5Q1,index9FG); index3aMod5Q1 = Mod5Q1; index3bMod5Q1 = Mod5Q1; index3cMod5Q1 = Mod5Q1; Index(Mod5Q1,index3dMod5Q1) = 3; index9Mod5Q1 = index3dMod5Q1; ##### # # Consider abelianizations # ##### # (C2,p=2,\{3\}) AbelianInvariants(GammaBarFP) = [3,3]; AbelianInvariants(GammaBarPSAMod2) = [3,3]; AbelianInvariants(Mod3Q1) = []; AbelianInvariants(Mod5Q1) = [3]; # (C2,p=2,\{3\},D_3) AbelianInvariants(index3aFP) = [3,7]; AbelianInvariants(index3aPSAMod2) = [3,7]; AbelianInvariants(index3aMod3Q1) = []; AbelianInvariants(index3aMod5Q1) = [3]; # (C2,p=2,\{3\},(d^2D)_3) AbelianInvariants(index3bFP) = [3]; AbelianInvariants(index3bPSAMod2) = [3]; AbelianInvariants(index3bMod3Q1) = []; AbelianInvariants(index3bMod5Q1) = [3]; # (C2,p=2,\{3\},(dD)_3) AbelianInvariants(index3cFP) = [3]; AbelianInvariants(index3cPSAMod2) = [3]; AbelianInvariants(index3cMod3Q1) = []; AbelianInvariants(index3cMod5Q1) = [3]; # (C2,p=2,\{3\},d_3) AbelianInvariants(index3dFP) = [3]; AbelianInvariants(index3dPSAMod2) = [3]; AbelianInvariants(index3dMod3Q1) = []; AbelianInvariants(index3dMod5Q1) = []; # (C2,p=2,\{3\},d_3 D_3) AbelianInvariants(index9FP) = [7]; AbelianInvariants(index9PSAMod2) = [7]; AbelianInvariants(index9Mod3Q1) = []; AbelianInvariants(index9Mod5Q1) = [];