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#  File converted from Magma code -- requires editing and checking
#  Magma -> GAP converter, version 0.5, 11/5/18 by AH

#  ///////////////////////////////////////////////////////////////////////
#     standard presentation for SU(n, q)                                //
#                                                                       //
#     Input two parameters to compute this presentation of SU(n, q)     //
#       n = dimension                                                   //
#       q = field order                                                 //
#                                                                       //
#     July 2018                                                         //
#  ///////////////////////////////////////////////////////////////////////

#   return presentation generators as words in standard generators
BindGlobal("SUStandardToPresentation@",
function(d,q)
local lvarDelta,lvarGamma,P,S,U,V,W,varZ,delta,rest,s,sigma,t,tau,v,x,y;
  W:=FreeGroup("Z","V","tau","delta","U","sigma","Delta");
  W:=Group(StraightLineProgGens(GeneratorsOfGroup(W)));
  S:=GeneratorsOfGroup(W);
  if IsEvenInt(d) then
    varZ:=S[1];
    varZ:=varZ^-1;
    V:=S[2];
    tau:=S[3];
    tau:=tau^-1;
    delta:=S[4];
    delta:=delta^-1;
    U:=S[5];
    sigma:=S[6];
    sigma:=sigma^(V^2);
    lvarDelta:=S[7]^(V^2);
    lvarDelta:=lvarDelta^-1;
    P:=[varZ,V,tau,delta,U,sigma,lvarDelta];
  else
    if d=3 then
      rest:=List([1..3],i->One(W));
      #   sequence [v, tau, Delta, t]
      if IsOddInt(q) then
        return Concatenation([S[6],S[3],S[7],(S[7]^-1)^(QuoInt((q+1),2))*S[1]]
         ,rest);
      else
        return Concatenation([S[6],S[3],S[7],S[1]],rest);
      fi;
    fi;
    y:=S[7];
    V:=S[2];
    U:=S[5];
    tau:=S[3];
    x:=S[6];
    s:=S[1];
    lvarGamma:=(y^V)^-1;
    v:=x^V;
    # rewritten select statement
    if IsEvenInt(q) then
      t:=s;
    else
      t:=lvarGamma^(QuoInt((q^2+q),2))*s^-1;
    fi;
    sigma:=Comm(v^(t*U),v^t)^t;
    #   return [Gamma, t, U, V, sigma, tau, v]
    P:=[lvarGamma,t,U,V,sigma,tau^-1,x^V];
  fi;
  return P;
end);

#   return standard generators as sequence [s, V, t, delta, U, x, y]
BindGlobal("SUPresentationToStandard@",
function(d,q)
local lvarDelta,lvarGamma,P,S,U,V,W,varZ,delta,sigma,t,tau,v;
  if d=3 then
    W:=FreeGroup(4);
    W:=Group(StraightLineProgGens(GeneratorsOfGroup(W)));
    P:=GeneratorsOfGroup(W);
    if IsOddInt(q) then
      return [P[3]^(QuoInt((q+1),2))*P[4],P[1]^0,P[2],P[3]^(q+1),P[1]^0,P[1]
       ,P[3]];
    else
      return [P[4],P[4]^0,P[2],P[3]^(q+1),P[3]^0,P[1],P[3]];
    fi;
  fi;
  W:=FreeGroup("Z","V","tau","delta","U","sigma","Delta");
  W:=Group(StraightLineProgGens(GeneratorsOfGroup(W)));
  P:=GeneratorsOfGroup(W);
  if IsEvenInt(d) then
    varZ:=P[1];
    varZ:=varZ^-1;
    V:=P[2];
    tau:=P[3];
    tau:=tau^-1;
    delta:=P[4];
    delta:=delta^-1;
    U:=P[5];
    sigma:=P[6];
    sigma:=sigma^(V^-2);
    lvarDelta:=P[7]^(V^-2);
    lvarDelta:=lvarDelta^-1;
    S:=[varZ,V,tau,delta,U,sigma,lvarDelta];
  else
    lvarGamma:=P[1];
    t:=P[2];
    U:=P[3];
    V:=P[4];
    sigma:=P[5];
    tau:=P[6];
    v:=P[7];
    # rewritten select statement
    if IsEvenInt(q) then
      varZ:=t;
    else
      varZ:=(lvarGamma^-1)^(QuoInt((q^2+q),2))*t;
    fi;
    S:=[varZ^-1,V,tau^-1,lvarGamma^-(q+1),U,v^(V^-1),(lvarGamma^-1)^(V^-1)];
  fi;
  return S;
end);

#   relations are on presentation generators;
#  convert to relations on standard generators
BindGlobal("SUConvertToStandard@",
function(d,q,Rels)
local A,B,C,T,U,W,tau,gens;
  if d=3 and q=2 then
    #return Universe(Rels)/Rels;
    return FamilyObj(Rels)!.wholeGroup/Rels;
  fi;
  A:=SUStandardToPresentation@(d,q);
  # was "Rels:=Evaluate(Rels,A);"
  gens:=GeneratorsOfGroup(FamilyObj(Rels)!.wholeGroup);
  Rels:=List(Rels, w-> MappedWord(w, gens, A));
  B:=SUPresentationToStandard@(d,q);
  # was "C:=Evaluate(B,A);"
  gens:=GeneratorsOfGroup(FamilyObj(B)!.wholeGroup);
  C:=List(B, w-> MappedWord(w, gens, A));
  U:=FamilyObj(C)!.wholeGroup;
  W:=FamilyObj(Rels)!.wholeGroup;

  tau:=GroupHomomorphismByImages(U,W,GeneratorsOfGroup(U),GeneratorsOfGroup(W));
  T:=List([1..Size(GeneratorsOfGroup(W))],
    i->W.(i)^-1*ImagesRepresentative(tau,C[i]));
  Rels:=Concatenation(Rels,T);
  return W/Rels;
end);

#   construct generator of centre of SU(d, q) as SLP in presentation generators
BindGlobal("SUGeneratorOfCentre@",
function(d,q,F)
local lvarDelta,lvarGamma,U,V,varZ,a,b,n,t,z;
  if Gcd(d,q+1)=1 then
    return One(F);
  fi;
  n:=QuoInt(d,2);
  if IsOddInt(d) then
    if d=3 then
      z:=F.3^(QuoInt((q^2-1),3));
      return z;
    fi;
    lvarGamma:=F.1;
    t:=F.2;
    V:=F.4;
    # rewritten select statement
    if IsEvenInt(q) then
      varZ:=t;
    else
      varZ:=t*lvarGamma^(QuoInt((q+1),2));
    fi;
    z:=(lvarGamma*varZ*V)^(QuoInt(2*n*(q+1),Gcd(q+1,2*n+1)));
  else
    varZ:=F.1;
    U:=F.5;
    V:=F.2;
    lvarDelta:=F.7;
    #a:=Valuation(q+1,2);
    a:=1;
    while IsInt((q+1)/(2*a)) do a:=a*2;od;

    #b:=Valuation(n,2);
    b:=1;
    while IsInt(n/(2*b)) do b:=b*2;od;

    if a > b then
      z:=varZ^2*(lvarDelta^(q-1)*U*V^-1)^(QuoInt((n-1)*(q+1),(2*Gcd(q+1,n))));
    else
      z:=(lvarDelta^(q-1)*U*V^-1)^(QuoInt((n-1)*(q+1),Gcd(q+1,n)));
    fi;
  fi;
  return z;
end);

InstallGlobalFunction(Internal_StandardPresentationForSU@,function(d,K)
#  -> ,GrpSLP ,[ ,]  return standard presentation for SU ( d , q ) ; if
#  Projective := true , then return presentation for PSU ( d , q )
local P,Presentation,Projective,R,Rels,S,z,q;
  Projective:=ValueOption("Projective");
  if Projective=fail then
    Projective:=false;
  fi;
  Presentation:=ValueOption("Presentation");
  if Presentation=fail then
    Presentation:=false;
  fi;

  if IsInt(K) then
    if not IsPrimePowerInt(K) then
      Error("<q> must be a prime power");
    fi;
    q:=K;
    K:=GF(q);
  else
    if not IsField(K) and IsFinite(K) then 
      Error("<K> must be a finite field");
    fi;
    q:=Size(K);
  fi;

  if not d > 2 then
    Error("Degree must be at least 3");
  fi;
  if Size(DuplicateFreeList(Factors(q))) > 1 then
    Error("Field size is not valid");
  fi;
  if d=3 and q=2 then
    return SU32Presentation@(:Projective:=Projective);
  elif IsOddInt(d) then
    R:=OddSUPresentation@(d,q:Projective:=false);
    P:=FreeGroupOfFpGroup(R);
    R:=RelatorsOfFpGroup(R);
  else
    R:=EvenSUPresentation@(d,q:Projective:=false);
    P:=FreeGroupOfFpGroup(R);
    R:=RelatorsOfFpGroup(R);
  fi;
  if Projective and Gcd(d,q+1) > 1 then
    z:=SUGeneratorOfCentre@(d,q,P);
    R:=Concatenation(R,[z]);
  fi;
  if Presentation then
    return P/R;
  fi;
  Rels:=SUConvertToStandard@(d,q,R);
  S:=FreeGroupOfFpGroup(Rels);
  Rels:=RelatorsOfFpGroup(Rels);
  Rels:=Filtered(Rels,w->not IsOne(w));
  return S/Rels;
end);

InstallGlobalFunction(SUGenerators@,function(d,q)
if d=3 then
    return SU3Generators@(q);
  elif IsOddInt(d) then
    return OddSUGenerators@(d,q);
  else
    return EvenSUGenerators@(d,q);
  fi;
end);