Improve readability and fix minor output bug

Author: Till Müller

gap/projective/constructive_recognition/SL/BaseCase.gi

@@ -522,6 +522,8 @@ end;
 ##      nicegens:=List(l[1],a->ResultOfStraightLineProgram(a,GeneratorsOfGroup(G)));
 ##      List(last, a->a^l[2]);
 ##  
+
+## Note does not work for q = 2,3,5
 RecogNaturalSL2 := function(G, q)
   local GM, one, zero, qm1fac, c, m, gens, xm, x, pol, v, z, exp, a, 
         mat, tm, ym, y, ymat, tr, d, cm, r1, r2, r, log, i, trupm, 
@@ -707,74 +709,75 @@ end;
 ## G must be SL(2,q) generated by 2x2 matrices over GF(q).
 ## Note: does not work for q = 2, 3, 5.
 ConRecogNaturalSL2 := function(G, f)
-local q, res, nicegens, diag, u1, u2, umat, lmat, k, j, i, el, result, bas, true_diag, true_u1, true_u2, basi, p, basis, coeffs, m, c, l;
-q := Size(f);
-p := Characteristic(f);
-j := DegreeOverPrimeField(GF(q));
-## standard generators of SL(2,q) in the natural representation
-true_diag  := [[Z(q)^(-1),0*Z(q)],[0*Z(q),Z(q)]];
-true_u1 := [[Z(q)^0,0*Z(q)],[Z(q)^0,Z(q)^0]];
-true_u2 := [[Z(q)^0,Z(q)^0],[0*Z(q),Z(q)^0]];
-## retry until RecogNaturalSL2 returns generators matching the standard form
-for i in [1..100] do
-res := RecogNaturalSL2(G,q);
-nicegens :=List(res[1],a->ResultOfStraightLineProgram(a,GeneratorsOfGroup(G)));
-diag := nicegens[1];
-u1 := nicegens[2];
-u2 := nicegens[3];
-## check if we found the correct generators
-if diag^res[2] = true_diag and u1^res[2] = true_u1 and u2^res[2] = true_u2 then 
-break; 
-fi;
-od;
-
-if IsEvenInt(q) then
-## even characteristic: conjugation by diag generates all of GF(q)* directly
-lmat := [];
-for k in [0..j-1] do
-    i := (q-1-k)*Int(2)^-1 mod (q-1);
-    el := u1^(diag^i);
-Add(lmat,el);
-od;
-umat := [];
-for k in [0..j-1] do
-    i := k * Int(2)^-1 mod (q-1);
-    el := u2^(diag^i);
-Add(umat, el);
-od;
-else
-## odd characteristic: conjugation only yields squares, so express z^l
-## in the Fp-basis {z^0, z^2, ..., z^(2(j-1))} and multiply conjugates
-basis := List([0..j-1], i -> Z(q)^(2*i));
-lmat := [];
-for l in [0..j-1] do
-  coeffs := Coefficients(Basis(GF(q), basis), Z(q)^l);
-  m := u1^0;
-for i in [0..j-1] do
-    c := IntFFE(coeffs[i+1]);
-    m := m * (diag^i * u1 * diag^(-i))^c;
-od;
-Add(lmat, m);
-od;
-umat := [];
-for l in [0..j-1] do
-  coeffs := Coefficients(Basis(GF(q), basis), Z(q)^l);
-  m := u2^0;
-for i in [0..j-1] do
-    c := IntFFE(coeffs[i+1]);
-    m := m * (diag^(-i) * u2 * diag^i)^c;
-od;
-Add(umat, m);
-od;
-fi;
-basi := res[2];
-bas := basi^(-1);
-result :=  rec( g := G, t := lmat, s := umat, bas := bas, basi := basi,
-              one := One(f), a := umat[1]*lmat[1]*umat[1], b := One(umat[1]),
-One := One(umat[1]), f := f, q := q, p := Characteristic(f), ext := j,
-              d := 2 );
-  result.all := Concatenation(result.s,result.t,[result.a],[result.b]);
-return result;
+  local q, res, nicegens, diag, u1, u2, umat, lmat, k, j, i, el, result, bas, true_diag, true_u1, true_u2, basi, p, basis, coeffs, m, c, l;
+  q := Size(f);
+  p := Characteristic(f);
+  j := DegreeOverPrimeField(GF(q));
+  ## standard generators of SL(2,q) in the natural representation
+  true_diag  := [[Z(q)^(-1),0*Z(q)],[0*Z(q),Z(q)]];
+  true_u1 := [[Z(q)^0,0*Z(q)],[Z(q)^0,Z(q)^0]];
+  true_u2 := [[Z(q)^0,Z(q)^0],[0*Z(q),Z(q)^0]];
+  ## retry until RecogNaturalSL2 returns generators matching the standard form
+  for i in [1..100] do
+    res := RecogNaturalSL2(G,q);
+    nicegens :=List(res[1],a->ResultOfStraightLineProgram(a,GeneratorsOfGroup(G)));
+    diag := nicegens[1];
+    u1 := nicegens[2];
+    u2 := nicegens[3];
+    ## check if we found the correct generators
+    if diag^res[2] = true_diag and u1^res[2] = true_u1 and u2^res[2] = true_u2 then 
+      break; 
+    fi;
+  od;
+
+  if IsEvenInt(q) then
+    ## even characteristic: conjugation by diag generates all of GF(q)* directly
+    lmat := [];
+    for k in [0..j-1] do
+      i := (q-1-k)*Int(2)^-1 mod (q-1);
+      el := u1^(diag^i);
+      Add(lmat,el);
+    od;
+    umat := [];
+    for k in [0..j-1] do
+      i := k * Int(2)^-1 mod (q-1);
+      el := u2^(diag^i);
+      Add(umat, el);
+    od;
+  else
+    ## odd characteristic: conjugation only yields squares, so express z^l
+    ## in the Fp-basis {z^0, z^2, ..., z^(2(j-1))} and multiply conjugates
+    basis := List([0..j-1], i -> Z(q)^(2*i));
+    lmat := [];
+    for l in [0..j-1] do
+      coeffs := Coefficients(Basis(GF(q), basis), Z(q)^l);
+      m := u1^0;
+      for i in [0..j-1] do
+        c := IntFFE(coeffs[i+1]);
+        m := m * (diag^i * u1 * diag^(-i))^c;
+      od;
+      Add(lmat, m);
+    od;
+    umat := [];
+    for l in [0..j-1] do
+      coeffs := Coefficients(Basis(GF(q), basis), Z(q)^l);
+      m := u2^0;
+      for i in [0..j-1] do
+        c := IntFFE(coeffs[i+1]);
+        m := m * (diag^(-i) * u2 * diag^i)^c;
+      od;
+      Add(umat, m);
+    od;
+  fi;
+  basi := res[2];
+  bas := basi^(-1);
+  a := umat[1]^(-1)*lmat[1]*umat[1]^(-1);
+  b := One(umat[1]);
+  result :=  rec( g := G, t := lmat, s := umat, bas := bas, basi := basi,
+                one := One(f), a := a, b := b, 
+                all := Concatenation(umat,lmat,[a],[b]), One := One(umat[1]), f := f,
+                q := q, p := Characteristic(f), ext := j, d := 2 );
+  return result;
 end;
 
 
@@ -783,43 +786,44 @@ end;
 ## Input: either a list of prime powers or the empty list 
 ## (then a preset list of prime powers is tested).
 test_ConRecogNaturalSL2 := function(input)
-local i, G, list, qlist, res_old, res, f, q, valid;
-if input = [] then
+  local i, G, list, qlist, res_old, res, f, q, valid;
+  if input = [] then
     qlist := [2^3, 2^5, 3^4, 25, 17^3, 9967]; 
-elif Size(input)>0 then
+  elif Size(input)>0 then
     qlist := [];
-for q in input do
-if IsPrimePowerInt(q) then
-Add(qlist, q);
-fi;
-od;
-fi;
-valid := true;
-for q in qlist do
-Print("testing q = ", q, "\n");
+    for q in input do
+      if IsPrimePowerInt(q) then
+        Add(qlist, q);
+      fi;
+    od;
+  fi;
+  
+  valid := true;
+  for q in qlist do
+    Print("testing q = ", q, "\n");
     f := GF(q);
     list := [];
-for i in [1..5] do
-Add(list, Random(SL(2,q)));
-od;
+    for i in [1..5] do
+      Add(list, Random(SL(2,q)));
+    od;
     G := GroupWithGenerators(list);
-if IsEvenInt(q) then
-        res_old := RECOG.RecogniseSL2NaturalEvenChar(G,f,false);
-else
-        res_old := RECOG.RecogniseSL2NaturalOddCharUsingBSGS(G,f);
-fi;
+    if IsEvenInt(q) then
+      res_old := RECOG.RecogniseSL2NaturalEvenChar(G,f,false);
+    else
+      res_old := RECOG.RecogniseSL2NaturalOddCharUsingBSGS(G,f);
+    fi;
     res := ConRecogNaturalSL2(G,f);
-## compare all generators after change of basis
-for i in [1..Length(res.all)] do
-if res.all[i]^res.basi <> res_old.all[i]^res_old.basi then
-Print("Test failed for q = ", q, ", index i = ", i, "\n");
+    ## compare all generators after change of basis
+    for i in [1..Length(res.all)] do
+      if res.all[i]^res.basi <> res_old.all[i]^res_old.basi then
+        Print("Test failed for q = ", q, ", index i = ", i, " in the list \"all\" failed\n");
             valid := false;
-fi;
-od;
-od;
-if valid then
-Print("All tests passed.\n");
-fi;
+      fi;
+    od;
+  od;
+  if valid then
+    Print("All tests passed.\n");
+  fi;
 end;