##### # # This is GAP code # # The file is ~steger/donald/SU21/a7p2N/a7p2N-0-proj.gap # # See file ~steger/donald/SU21/a7p2N/D.txt for the definition of the # division algebra $\D$ and for all the notation. # ##### Read("/home/steger/donald/SU21/a7p2N/a7p2N-proj.gap"); Read("/home/steger/donald/SU21/NormElem.gap"); ##### # # Free group # ###### #From: "Donald Cartwright" #To: steger@uniss.it #Subject: Re: Elements of finite order #Date: Tue, 15 Jan 2008 20:22:26 +1100 FreeGp:=FreeGroup("Z","B","C"); Zf:=FreeGp.1; Bf:=FreeGp.2; Cf:=FreeGp.3; RelnList:=[ Bf^3, Zf^7, (Bf*Zf^2*Cf^2)^3, (Zf^-1*Cf^2*Zf^-1*Bf^-1*Zf^-1*Bf^-1*Zf^-1)^3, Zf^2*Bf*Zf*Bf*Zf^-1*Cf^-2*Zf^-2*Bf, Cf*Zf^-2*Bf*Zf*Cf^-1*Zf^-2*Bf^-1*Zf^3, Zf^2*Cf^-1*Zf^-1*Bf^-1*Zf^2*Cf*Zf^-1*Bf^-1*Zf^-1*Bf^-1, Bf^-1*Zf^-3*Cf^-1*Zf^-2*Cf^-1*Zf^-2*Bf*Zf^-1, Zf*Cf*Zf^-2*Bf*Zf^-1*Bf*Zf^-1*Bf*Zf^2*Cf, Zf^2*Cf^-1*Zf^-2*Bf*Zf^-1*Cf^-2*Bf*Zf*Bf*Zf^-1*Cf^-1, Zf^-2*Cf^-1*Zf*Cf*Zf^-2*Bf*Zf^-1*Cf^-1*Zf^-2*Bf^-1*Zf^-1*Cf^-1, Zf^-2*Bf*Zf^-1*Bf^-1*Zf^-1*Bf^-1*Zf^2*Bf*Zf^-1*Bf*Zf^2*Cf^2*Zf^-1*Bf^-1, Cf*Bf*Zf^2*Cf^2*Bf^-1*Zf^-1*Cf^-2*Zf^-3*Bf^-1*Cf*Zf^-2*Bf*Zf, Zf^-1*Bf^-1*Zf^-1*Bf^-1*Zf^3*Bf^-1*Cf^-1*Zf^-2*Bf^-1*Zf^-2*Cf*Zf^3*Bf ]; GammaBarFP:=FreeGp/RelnList;; ZFP:=GammaBarFP.1; BFP:=GammaBarFP.2; CFP:=GammaBarFP.3; #From: "Donald Cartwright" #To: "steger@uniss.it" #Subject: Group (7,2), new setup <--> old setup #Date: Sat, 19 Jan 2008 03:11:13 +1100 # # As explained in this message ``DFP'' corresponds to what was called # ``B'' in the old setup. UFP:=ZFP^2; DFP:=CFP*ZFP^3*BFP*ZFP^2; Group(UFP,DFP)=GammaBarFP; # Important subgroups of GammaBarFP index3FP:=Group(UFP, DFP*UFP*DFP^-1, DFP^3); index7aFP:=Group(DFP, UFP^2*DFP^-1*UFP, UFP*DFP*UFP^2); index7bFP:=Group(DFP, UFP^2*DFP*UFP^-1, UFP^-2*DFP*UFP); index21aFP:=Group(DFP^3, DFP^-1*UFP^-1*DFP*UFP^-2, DFP*UFP^-2*DFP^-1*UFP^-1, UFP*DFP^3*UFP, UFP*DFP^-1*UFP*DFP*UFP, UFP^-1*DFP^3*UFP^-1); index21bFP:=Group(DFP^3, DFP*UFP*DFP^-1*UFP^-2, UFP*DFP^3*UFP^-1, UFP^-1*DFP*UFP*DFP^-1*UFP^-1, DFP^-1*UFP^2*DFP*UFP^-1, UFP^2*DFP^-2*UFP^-1*DFP^-1); index21cFP:=Group(DFP, UFP*DFP^-2*UFP, UFP^2*DFP*UFP^2, UFP^3*DFP^-1*UFP^2); index3FCA:=FactorCosetAction(GammaBarFP,index3FP); index7aFCA:=FactorCosetAction(GammaBarFP,index7aFP); index7bFCA:=FactorCosetAction(GammaBarFP,index7bFP); index21aFCA:=FactorCosetAction(GammaBarFP,index21aFP); index21bFCA:=FactorCosetAction(GammaBarFP,index21bFP); index21cFCA:=FactorCosetAction(GammaBarFP,index21cFP); # Check the indexes of the subgroups Index(GammaBarFP,index3FP) = 3; Index(GammaBarFP,index7aFP) = 7; Index(GammaBarFP,index7bFP) = 7; Index(GammaBarFP,index21aFP) = 21; Index(GammaBarFP,index21bFP) = 21; Index(GammaBarFP,index21cFP) = 21; # Check normality IsNormal(GammaBarFP,index3FP); not IsNormal(GammaBarFP,index7aFP); not IsNormal(GammaBarFP,index7bFP); not IsNormal(GammaBarFP,index21aFP); IsNormal(GammaBarFP,index21bFP); not IsNormal(GammaBarFP,index21cFP); # Free group versions of these subgroups FPtoFGSubgp:=g->Group(List(GeneratorsOfGroup(g),UnderlyingElement)); index3FG:=FPtoFGSubgp(index3FP); index7aFG:=FPtoFGSubgp(index7aFP); index7bFG:=FPtoFGSubgp(index7bFP); index21aFG:=FPtoFGSubgp(index21aFP); index21bFG:=FPtoFGSubgp(index21bFP); index21cFG:=FPtoFGSubgp(index21cFP); ##### # # Generators of $(a=7,p=2,\emptyset)$ # ##### #From: "Donald Cartwright" #To: "steger@uniss.it" #Subject: Three possible generators in the new set up of group (7,2) #Date: Tue, 8 Jan 2008 23:44:00 +1100 ZDM=[[Zm, 0, 0], [0,Zm^2, 0], [0, 0,Zm^4]]; BDM:=[[1/7*(6*Zm^4-3*Zm^3-6*Zm^2-3*Zm-1), 1/7*(6*Zm^5+2*Zm^4+2*Zm^3+6*Zm^2+7*Zm+5), 1/7*(-2*Zm^5-6*Zm^4-5*Zm^3+Zm^2-2*Zm-7)], [ 1/7*(2*Zm^5+Zm^4-3*Zm^3-3*Zm^2-6*Zm-5), 1/7*(3*Zm^5-3*Zm^4+3*Zm^3+9*Zm+2), 1/7*(-2*Zm^5+4*Zm^4+4*Zm^3+5*Zm^2+3)], [ 1/14*(-6*Zm^5-2*Zm^4-9*Zm^3-6*Zm^2+7*Zm-5), 1/7*(3*Zm^5+5*Zm^3-3*Zm^2+4*Zm-2), 1/7*(-3*Zm^5-3*Zm^4+6*Zm^2-6*Zm-1)]]; CDM:=[[ 1/7*(5*Zm^5+9*Zm^4+5*Zm^3+7*Zm^2+Zm+1), 1/7*(5*Zm^5+14*Zm^4+6*Zm^3+2*Zm^2+9*Zm+13), 1/7*(4*Zm^5+13*Zm^4+13*Zm^3+4*Zm^2+7*Zm+8)], [ 1/7*(-8*Zm^5-5*Zm^4+2*Zm^3-Zm^2+7*Zm-2), 1/7*(-5*Zm^5+2*Zm^4-4*Zm^2+4*Zm-4), 1/7*(-6*Zm^5-4*Zm^4-Zm^3+3*Zm^2+8*Zm+7)], [ 1/7*(-8*Zm^5+3*Zm^4-9*Zm^3+5*Zm^2-4*Zm-1), 1/7*(-2*Zm^5-3*Zm^4-10*Zm^3+5*Zm^2-7*Zm-4), 1/7*(-4*Zm^4-5*Zm^3+4*Zm^2+2*Zm-4)]]; ZDM in DM; BDM in DM; CDM in DM; IsOne(iotaDM(ZDM)*ZDM); IsOne(iotaDM(BDM)*BDM); IsOne(iotaDM(CDM)*CDM); IsZero(LieBracket(iotaPDM,ZPDM)); IsZero(LieBracket(iotaPDM,BPDM)); IsZero(LieBracket(iotaPDM,CPDM)); # Check that the relations hold (projectively) for all our matrix # generators GammaBarDM:=Group(ZDM,BDM,CDM);; DMHom:=GroupHomomorphismByImages(FreeGp,GammaBarDM, [Zf,Bf,Cf],[ZDM,BDM,CDM]);; 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);; BPDM:=DMtoPDM(BDM);; CPDM:=DMtoPDM(CDM);; GammaBarPDM:=Group(ZPDM,BPDM,CPDM);; FGtoPDM:=GroupHomomorphismByImages(FreeGp,GammaBarPDM, [Zf,Bf,Cf],[ZPDM,BPDM,CPDM]);; 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);; BPSA0:=PDMtoPSA(BPDM);; CPSA0:=PDMtoPSA(CPDM);; ##### # # Adjust the basis of the self-adjoint part of~$\D$ so that # the group generators correspond to integral matrices. # ##### GammaBarPSA0:=Group(ZPSA0,BPSA0,CPSA0);; ILatPSA0:=InvariantLattice(GammaBarPSA0)^-1; ZPSA:=ZPSA0^ILatPSA0; BPSA:=BPSA0^ILatPSA0; CPSA:=CPSA0^ILatPSA0; GammaBarPSA:=Group(ZPSA,BPSA,CPSA);; FGtoPSA:=GroupHomomorphismByImages(FreeGp,GammaBarPSA, [Zf,Bf,Cf],[ZPSA,BPSA,CPSA]);; ForAll(RelnList,x->IsOne(x^FGtoPSA)); ###### # # Do calculations $\mod 2$ # ###### toMod2:=x->x*One(GF(2)); toMod2M:=mtx->fOnM(toMod2,mtx); ZPSAMod2:=toMod2M(ZPSA);; BPSAMod2:=toMod2M(BPSA);; CPSAMod2:=toMod2M(CPSA);; GammaBarPSAMod2:=Group(ZPSAMod2,BPSAMod2,CPSAMod2);; FGtoPSAMod2:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod2, [Zf,Bf,Cf],[ZPSAMod2,BPSAMod2,CPSAMod2]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod2)); #Size(GammaBarPSAMod2) = 21*2^6; #StructureDescription(GammaBarPSAMod2) # = "(((C2 x C2 x C2) . (C2 x C2 x C2)) : C7) : C3"; index3PSAMod2:=Image(FGtoPSAMod2,index3FG); index7aPSAMod2:=Image(FGtoPSAMod2,index7aFG); index7bPSAMod2:=Image(FGtoPSAMod2,index7bFG); index21aPSAMod2:=Image(FGtoPSAMod2,index21aFG); index21bPSAMod2:=Image(FGtoPSAMod2,index21bFG); index21cPSAMod2:=Image(FGtoPSAMod2,index21cFG); Index(GammaBarPSAMod2,index3PSAMod2) = 3; index7aPSAMod2 = GammaBarPSAMod2; Index(GammaBarPSAMod2,index7bPSAMod2) = 7; index21aPSAMod2 = index3PSAMod2; Index(GammaBarPSAMod2,index21bPSAMod2) = 21; #StructureDescription(index21bPSAMod2) = # "(C2 x C2 x C2) . (C2 x C2 x C2)"; index21cPSAMod2 = GammaBarPSAMod2; ###### # # Do calculations $\mod 3$ # ###### toMod3:=x->x*One(GF(3)); toMod3M:=mtx->fOnM(toMod3,mtx); ZPSAMod3:=toMod3M(ZPSA);; BPSAMod3:=toMod3M(BPSA);; CPSAMod3:=toMod3M(CPSA);; GammaBarPSAMod3:=Group(ZPSAMod3,BPSAMod3,CPSAMod3);; FGtoPSAMod3:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod3, [Zf,Bf,Cf],[ZPSAMod3,BPSAMod3,CPSAMod3]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod3)); Size(GammaBarPSAMod3) = 6048; StructureDescription(GammaBarPSAMod3) = "PSU(3,3)"; index3PSAMod3:=Image(FGtoPSAMod3,index3FG); index7aPSAMod3:=Image(FGtoPSAMod3,index7aFG); index7bPSAMod3:=Image(FGtoPSAMod3,index7bFG); index21aPSAMod3:=Image(FGtoPSAMod3,index21aFG); index21bPSAMod3:=Image(FGtoPSAMod3,index21bFG); index21cPSAMod3:=Image(FGtoPSAMod3,index21cFG); index3PSAMod3 = GammaBarPSAMod3; index7aPSAMod3 = GammaBarPSAMod3; index7bPSAMod3 = GammaBarPSAMod3; index21aPSAMod3 = GammaBarPSAMod3; index21bPSAMod3 = GammaBarPSAMod3; index21cPSAMod3 = GammaBarPSAMod3; ###### # # Do calculations $\mod 5$ # ###### toMod5:=x->x*One(GF(5)); toMod5M:=mtx->fOnM(toMod5,mtx); ZPSAMod5:=toMod5M(ZPSA);; BPSAMod5:=toMod5M(BPSA);; CPSAMod5:=toMod5M(CPSA);; GammaBarPSAMod5:=Group(ZPSAMod5,BPSAMod5,CPSAMod5);; FGtoPSAMod5:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod5, [Zf,Bf,Cf],[ZPSAMod5,BPSAMod5,CPSAMod5]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod5)); CSGammaBarPSAMod5:=CompositionSeries(GammaBarPSAMod5); Length(CSGammaBarPSAMod5) = 3; StructureDescription(CSGammaBarPSAMod5[2]) = "PSU(3,5)"; Size(GammaBarPSAMod5) = 378000; GammaBarPSAMod5Hom3:=NaturalHomomorphismByNormalSubgroup (GammaBarPSAMod5,CSGammaBarPSAMod5[2]);; Size(Range(GammaBarPSAMod5Hom3)) = 3; #ForAny(ConjugacyClasses(GammaBarPSAMod5),c-> # Order(Representative(c))=3 # and Order(Representative(c)^GammaBarPSAMod5Hom3)=3); #StructureDescription(GammaBarPSAMod5) = "PSU(3,5) : C3"; index3PSAMod5:=Image(FGtoPSAMod5,index3FG); index7aPSAMod5:=Image(FGtoPSAMod5,index7aFG); index7bPSAMod5:=Image(FGtoPSAMod5,index7bFG); index21aPSAMod5:=Image(FGtoPSAMod5,index21aFG); index21bPSAMod5:=Image(FGtoPSAMod5,index21bFG); index21cPSAMod5:=Image(FGtoPSAMod5,index21cFG); Index(GammaBarPSAMod5,index3PSAMod5) = 3; index7aPSAMod5 = GammaBarPSAMod5; index7bPSAMod5 = GammaBarPSAMod5; index21aPSAMod5 = index3PSAMod5; index21bPSAMod5 = index3PSAMod5; index21cPSAMod5 = GammaBarPSAMod5; ###### # # Do calculations $\mod 7$ # ###### toMod7:=x->x*One(GF(7)); toMod7M:=mtx->fOnM(toMod7,mtx); ZPSAMod7:=toMod7M(ZPSA);; BPSAMod7:=toMod7M(BPSA);; CPSAMod7:=toMod7M(CPSA);; GammaBarPSAMod7:=Group(ZPSAMod7,BPSAMod7,CPSAMod7);; FGtoPSAMod7:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod7, [Zf,Bf,Cf],[ZPSAMod7,BPSAMod7,CPSAMod7]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod7)); #Size(GammaBarPSAMod7) = 5647152; 5647152 = 336*7^5; Mod7H1:=NormElem(GammaBarPSAMod7,7); Size(Mod7H1)=7^5; StructureDescription(Mod7H1)="C7 x C7 x C7 x C7 x C7"; Mod7ASA1:=AbelianSubfactorAction(GammaBarPSAMod7,Mod7H1,Group(One(Mod7H1))); Mod7Q1:=Image(Mod7ASA1[1]); Size(Mod7Q1)*Size(Mod7H1) = Size(GammaBarPSAMod7); IdSmallGroup(Mod7Q1) = IdSmallGroup(PGL(2,7)); #CSGammaBarPSAMod7:=CompositionSeries(GammaBarPSAMod7); #StructureDescription(GammaBarPSAMod7/CSGammaBarPSAMod7[3]) # = "PSL(3,2) : C2"; #StructureDescription(PGL(2,7)) = "PSL(3,2) : C2"; #Size(PGL(2,7)) = 336; #StructureDescription(CSGammaBarPSAMod7[3]) = "C7 x C7 x C7 x C7 x C7"; #ChSGammaBarPSAMod7:=ChiefSeries(GammaBarPSAMod7); #List(ChSGammaBarPSAMod7,h->Index(GammaBarPSAMod7,h)) # = [ 1, 2, 336, 5647152 ]; #PSAMod7Hom:=NaturalHomomorphismByNormalSubgroup( # GammaBarPSAMod7,ChSGammaBarPSAMod7[3]);; #StructureDescription(Group( # List(GeneratorsOfGroup(index21cPSAMod7),x->x^PSAMod7Hom) #)) = "D16"; index3PSAMod7:=Image(FGtoPSAMod7,index3FG); index7aPSAMod7:=Image(FGtoPSAMod7,index7aFG); index7bPSAMod7:=Image(FGtoPSAMod7,index7bFG); index21aPSAMod7:=Image(FGtoPSAMod7,index21aFG); index21bPSAMod7:=Image(FGtoPSAMod7,index21bFG); index21cPSAMod7:=Image(FGtoPSAMod7,index21cFG); index3PSAMod7 = GammaBarPSAMod7; index7aPSAMod7 = GammaBarPSAMod7; index7bPSAMod7 = GammaBarPSAMod7; index21aPSAMod7 = GammaBarPSAMod7; index21bPSAMod7 = GammaBarPSAMod7; Index(GammaBarPSAMod7,index21cPSAMod7) = 21; ####### ## ## Do calculations $\mod 49$ ## ####### # #toMod49M:=mtx->fOnM(x->ZmodnZObj(x,49),mtx); # ##GammaBarPSAMod49:=Group(List( ##[ [ [ 1, 47, 1, 0, 0, 0, 1, 48 ], [ 48, 48, 1, 0, 0, 0, 1, 0 ], ## [ 47, 45, 3, 0, 0, 0, 2, 48 ], [ 0, 0, 0, 1, 0, 0, 0, 0 ], ## [ 21, 42, 42, 3, 2, 5, 8, 46 ], [ 35, 0, 7, 47, 48, 46, 46, 1 ], ## [ 0, 0, 0, 0, 0, 0, 1, 0 ], [ 7, 42, 0, 1, 1, 2, 5, 46 ] ], ## [ [ 2, 19, 12, 46, 5, 4, 18, 39 ], [ 31, 37, 23, 43, 47, 38, 0, 41 ], ## [ 37, 48, 35, 40, 5, 45, 24, 29 ], [ 47, 14, 8, 4, 43, 36, 32, 46 ], ## [ 29, 28, 45, 46, 19, 33, 10, 36 ], [ 24, 3, 9, 40, 47, 43, 46, 4 ], ## [ 34, 7, 25, 0, 41, 27, 30, 40 ], [ 17, 40, 23, 0, 9, 6, 32, 25 ] ], ## [ [ 40, 0, 3, 0, 0, 48, 47, 1 ], [ 12, 30, 44, 9, 8, 15, 15, 45 ], ## [ 30, 42, 12, 0, 1, 47, 47, 48 ], [ 20, 33, 37, 11, 8, 17, 16, 46 ], ## [ 4, 15, 34, 39, 30, 22, 3, 7 ], [ 9, 18, 42, 25, 32, 7, 18, 45 ], ## [ 10, 34, 43, 9, 7, 14, 12, 47 ], [ 21, 7, 2, 1, 0, 0, 41, 8 ] ] ] ##,mtx->List(mtx,row->List(row,x->ZmodnZObj(x,49))))); # #ZPSAMod49:=toMod49M(ZPSA);; #BPSAMod49:=toMod49M(BPSA);; #CPSAMod49:=toMod49M(CPSA);; # #GammaBarPSAMod49:=Group(ZPSAMod49,BPSAMod49,CPSAMod49);; #FGtoPSAMod49:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod49, # [Zf,Bf,Cf],[ZPSAMod49,BPSAMod49,CPSAMod49]);; #ForAll(RelnList,x->IsOne(x^FGtoPSAMod49)); # #Mod49H1:=NormElem(GammaBarPSAMod49,7); #Size(Mod49H1)=7^5; #Mod49H1LAL:=LinearActionLayer( # GammaBarPSAMod49,GeneratorsOfGroup(GammaBarPSAMod49),Pcgs(Mod49H1)); #Mod49H1Q1:=Group(Mod49H1LAL); #IdSmallGroup(Mod49H1Q1) = IdSmallGroup(PGL(2,7)); # #MTXMod49H1:=GModuleByMats(Mod49H1LAL,GF(7)); #MTX.IsAbsolutelyIrreducible(MTXMod49H1); # #index3PSAMod49:=Image(FGtoPSAMod49,index3FG); #index7aPSAMod49:=Image(FGtoPSAMod49,index7aFG); #index7bPSAMod49:=Image(FGtoPSAMod49,index7bFG); #index21aPSAMod49:=Image(FGtoPSAMod49,index21aFG); #index21bPSAMod49:=Image(FGtoPSAMod49,index21bFG); #index21cPSAMod49:=Image(FGtoPSAMod49,index21cFG); # #index3PSAMod49 = GammaBarPSAMod49; #index7aPSAMod49 = GammaBarPSAMod49; #index7bPSAMod49 = GammaBarPSAMod49; #index21aPSAMod49 = GammaBarPSAMod49; #index21bPSAMod49 = GammaBarPSAMod49; #Index(GammaBarPSAMod49,index21cPSAMod49) = 21; ###### # # Do calculations $\mod 11$ # ###### toMod11:=x->x*One(GF(11)); toMod11M:=mtx->fOnM(toMod11,mtx); ZPSAMod11:=toMod11M(ZPSA);; BPSAMod11:=toMod11M(BPSA);; CPSAMod11:=toMod11M(CPSA);; GammaBarPSAMod11:=Group(ZPSAMod11,BPSAMod11,CPSAMod11);; FGtoPSAMod11:=GroupHomomorphismByImages(FreeGp,GammaBarPSAMod11, [Zf,Bf,Cf],[ZPSAMod11,BPSAMod11,CPSAMod11]);; ForAll(RelnList,x->IsOne(x^FGtoPSAMod11)); #IsSimple(GammaBarPSAMod11); #CSGammaBarPSAMod11:=CompositionSeries(GammaBarPSAMod11); #Size(GammaBarPSAMod11) = 212427600; #Size(PSL(3,11)) = 212427600; #StructureDescription(GammaBarPSAMod11) = "PSL(3,11)"; index3PSAMod11:=Image(FGtoPSAMod11,index3FG); index7aPSAMod11:=Image(FGtoPSAMod11,index7aFG); index7bPSAMod11:=Image(FGtoPSAMod11,index7bFG); index21aPSAMod11:=Image(FGtoPSAMod11,index21aFG); index21bPSAMod11:=Image(FGtoPSAMod11,index21bFG); index21cPSAMod11:=Image(FGtoPSAMod11,index21cFG); index3PSAMod11 = GammaBarPSAMod11; index7aPSAMod11 = GammaBarPSAMod11; index7bPSAMod11 = GammaBarPSAMod11; index21aPSAMod11 = GammaBarPSAMod11; index21bPSAMod11 = GammaBarPSAMod11; index21cPSAMod11 = GammaBarPSAMod11; ##### # # Consider abelianizations # ##### # (a=7,p=2,\emptyset) AbelianInvariants(GammaBarFP) = [2,3]; AbelianInvariants(GammaBarPSAMod2) = [3]; AbelianInvariants(GammaBarPSAMod7) = [2]; # (a=7,p=2,\emptyset,D_3) AbelianInvariants(index3FP) = [2,7]; AbelianInvariants(index3PSAMod2) = [7]; AbelianInvariants(index3PSAMod7) = [2]; # (a=7,p=2,\emptyset,X_7) AbelianInvariants(index7aFP) = [2,3]; AbelianInvariants(index7aPSAMod2) = [3]; AbelianInvariants(index7aPSAMod7) = [2]; # (a=7,p=2,\emptyset,2_7) AbelianInvariants(index7bFP) = [2,2,3]; AbelianInvariants(index7bPSAMod2) = [2,3]; AbelianInvariants(index7bPSAMod7) = [2]; # (a=7,p=2,\emptyset,D_3 X_7) AbelianInvariants(index21aFP) = [2,7]; AbelianInvariants(index21aPSAMod2) = [7]; AbelianInvariants(index21aPSAMod7) = [2]; # (a=7,p=2,\emptyset,D_3 2_7) AbelianInvariants(index21bFP) = [2,2,2,2]; AbelianInvariants(index21bPSAMod2) = [2,2,2]; AbelianInvariants(index21bPSAMod7) = [2]; # (a=7,p=2,\emptyset,7_{21}) AbelianInvariants(index21cFP) = [2,2,3,7]; AbelianInvariants(index21cPSAMod2) = [3]; AbelianInvariants(index21cPSAMod7) = [2,2,7];