CLEANUP: Replace calls to FactInt with Factors(Integers,

This way better factorizations methods in packages will get used.
(Calls in the recursive implementation of FactInt are kept.)
Also replace some calls by `PrimePGroup` etc.

(This is a revised version of earlier as there was already a merge with
PrimePGroup.)

This addresses #2087

Author: hulpke

lib/claspcgs.gi

@@ -606,7 +606,7 @@ local  G,  home,  # the group and the home pcgs
     cent:=false;
 
   elif IsPrimePowerInt(Size(G)) then
-    p:=FactorsInt(Size(G))[1];
+    p:=PrimePGroup(G);
     home:=PcgsPCentralSeriesPGroup(G);
     eas:=PCentralNormalSeriesByPcgsPGroup(home);
 
@@ -1107,7 +1107,7 @@ local  G,  home,  # the group and the home pcgs
         (InducedPcgs(home,cl.centralizer), c -> Comm(k, c) in L));
     end;
   elif IsPrimePowerInt(Size(G)) then
-    p:=FactorsInt(Size(G))[1];
+    p:=PrimePGroup(G);
     home:=PcgsPCentralSeriesPGroup(G);
     eas:=PCentralNormalSeriesByPcgsPGroup(home);
 

lib/ctbl.gi

@@ -1251,7 +1251,7 @@ InstallGlobalFunction( CharacterTable_IsNilpotentNormalSubgroup,
     orders:= OrdersClassRepresentatives( tbl );
     ppow:= Filtered( N, i -> IsPrimePowerInt( orders[i] ) );
 
-    for part in Collected( FactorsInt( Sum( classlengths{ N }, 0 ) ) ) do
+    for part in Collected( Factors(Integers, Sum( classlengths{ N }, 0 ) ) ) do
 
       # Check whether the Sylow p subgroup of `N' is normal in `N',
       # i.e., whether the number of elements of p-power is equal to
@@ -2903,7 +2903,7 @@ InstallMethod( PrimeBlocksOp,
         if d = ppart then
           d:= 0;
         else
-          d:= Length( FactorsInt( ppart / d ) );              # the defect
+          d:= Length( Factors(Integers, ppart / d ) );              # the defect
         fi;
         Add( primeblocks.defect, d );
 
@@ -4923,7 +4923,7 @@ BindGlobal( "CharacterTableDisplayDefault", function( tbl, options )
        elif centralizers = true then
           Print( "\n" );
           for i in [col..col+acol-1] do
-             fak:= FactorsInt( tbl_centralizers[classes[i]] );
+             fak:= Factors(Integers, tbl_centralizers[classes[i]] );
              for prime in Set( fak ) do
                 cen[prime][i]:= Number( fak, x -> x = prime );
              od;
@@ -5279,8 +5279,8 @@ InstallMethod( CharacterTableDirectProduct,
 
     # Compute power maps for all prime divisors of the result order.
     vals_direct:= ComputedPowerMaps( direct );
-    for k in Union( FactorsInt( Size( tbl1 ) ),
-                    FactorsInt( Size( tbl2 ) ) ) do
+    for k in Union( Factors(Integers, Size( tbl1 ) ),
+                    Factors(Integers, Size( tbl2 ) ) ) do
       powermap_k:= [];
       vals1:= PowerMap( tbl1, k );
       vals2:= PowerMap( tbl2, k );

lib/ctblfuns.gi

@@ -775,7 +775,7 @@ InstallMethod( CorrespondingPermutations,
           # Note that if we have taken away a union of orbits such that the
           # number of remaining points is smaller than the smallest prime
           # divisor of the order of `g' then all these points must be fixed.
-          min:= FactorsInt( Order( g ) )[1];
+          min:= Factors(Integers, Order( g ) )[1];
           images:= [];
 
           for list in part do
@@ -942,7 +942,7 @@ InstallOtherMethod( CorrespondingPermutations,
           # Note that if we have taken away a union of orbits such that the
           # number of remaining points is smaller than the smallest prime
           # divisor of the order of `g' then all these points must be fixed.
-          min:= FactorsInt( Order( g ) )[1];
+          min:= Factors(Integers, Order( g ) )[1];
           images:= [];
 
           for list in part do

lib/ctblmaps.gi

@@ -167,7 +167,7 @@ InstallOtherMethod( PowerMapOp,
     fi;
 
     image:= class;
-    for i in FactorsInt( n ) do
+    for i in Factors(Integers, n ) do
       # Here we use that `n' is a small integer.
       if not IsBound( powermap[i] ) then
 
@@ -3368,7 +3368,7 @@ InstallGlobalFunction( ConsiderSmallerPowerMaps, function( arg )
 
     for i in omega do
 
-      factors:= FactorsInt( prime mod tbl_orders[i] );
+      factors:= Factors(Integers, prime mod tbl_orders[i] );
       if factors = [ 1 ] or factors = [ 0 ] then factors:= []; fi;
 
       if ForAll( Set( factors ), x -> IsBound( tbl_powermap[x] ) ) then

lib/ctblmono.gi

@@ -939,7 +939,7 @@ InstallMethod( IsMonomialNumber,
           pair2,     # loop over `collect'
           ord;       # multiplicative order
 
-    factors := FactorsInt( n );
+    factors := Factors(Integers, n );
     collect := Collected( factors );
 
     # Get $\nu_2(n)$.
@@ -1106,7 +1106,7 @@ InstallMethod( TestMonomialQuick,
     if IsSolvableGroup( G ) then
 
       pi   := PrimeDivisors( codegree );
-      hall := Product( Filtered( FactorsInt( factsize ), x -> x in pi ), 1 );
+      hall := Product( Filtered( Factors(Integers, factsize ), x -> x in pi ), 1 );
 
       if factsize / hall = chi[1] then
 
@@ -1957,7 +1957,7 @@ InstallMethod( IsMinimalNonmonomial,
     factsize:= Index( K, F );
 
     # The Fitting subgroup of a minimal nomonomial group is a $p$-group.
-    facts:= FactorsInt( Size( F ) );
+    facts:= Factors(Integers, Size( F ) );
     p:= Set( facts );
     if 1 < Length( p ) then
       return false;

lib/ctblpope.gi

@@ -1888,7 +1888,7 @@ InstallGlobalFunction( PermCandidatesFaithful,
     od;
     # `primes': prime divisors of $|U|$ for which there is only one $G$-family
     # of that element order in $UN$:
-    factors:= FactorsInt( tbl_size / torso[1] );
+    factors:= Factors(Integers, tbl_size / torso[1] );
     primes:= Set( factors );
     orbits:= List( primes, p -> [] );
     for i in [ 1 .. nccl ] do

lib/cyclotom.gi

@@ -264,7 +264,7 @@ InstallGlobalFunction( CoeffsCyc, function( z, N )
         # must be equal, and the negative of this value is put at the
         # position of the $p$-th element of this congruence class.
         if second > 1 then
-          for p in FactorsInt( second ) do
+          for p in Factors(Integers, second ) do
             nn:= n / p;
             newcoeffs:= ListWithIdenticalEntries( nn, 0 );
             for k in [ 1 .. n ] do
@@ -1245,7 +1245,7 @@ InstallGlobalFunction( Quadratic, function( arg )
     fi;
 
     coeffs:= ExtRepOfObj( cyc );
-    facts:= FactorsInt( Length( coeffs ) );
+    facts:= Factors(Integers, Length( coeffs ) );
     factsset:= Set( facts );
     two_part:= Number( facts, x -> x = 2 );
 

lib/ffe.gi

@@ -819,7 +819,7 @@ InstallMethod( Order,
     p := Characteristic(z);
     d := DegreeFFE(z);
     ord := p^d-1;
-    facs := Collected(FactorsInt(ord));
+    facs := Collected(Factors(Integers,ord));
     for f in facs do
         for i in [1..f[2]] do
             o := ord/f[1];

lib/ffeconway.gi

@@ -518,7 +518,7 @@ FFECONWAY.WriteOverSmallestField := function(x)
         return x![3];
     fi;
     d := x![2];
-    f := Collected(FactorsInt(d));
+    f := Collected(Factors(Integers,d));
     for fac in f do
         l := fac[1];
         d1 := d/l;
@@ -1355,7 +1355,7 @@ FFECONWAY.DoLogFFE :=
     fi;
 
     # use rho method
-    f:=FactorsInt(q-1:quiet); # Quick factorization, don't stop if its too hard
+    f:=Factors(Integers,q-1:quiet); # Quick factorization, don't stop if its too hard
      return FFECONWAY.DoLogFFERho(y,z,q-1,f,q);
  end;
 
@@ -1392,7 +1392,7 @@ InstallMethod( Order,
     p := Characteristic(z);
     d := DegreeFFE(z);
     ord := p^d-1;
-    facs := Collected(FactorsInt(ord));
+    facs := Collected(Factors(Integers,ord));
     for f in facs do
         for i in [1..f[2]] do
             o := ord/f[1];

lib/fldabnum.gi

@@ -823,7 +823,7 @@ InstallGlobalFunction( ZumbroichBase, function( n, m )
       Error( "<m> must be a divisor of <n>" );
     fi;
 
-    factsn:= FactorsInt( n );
+    factsn:= Factors(Integers, n );
     primes:= Set( factsn );
     exponsn:= List( primes, x -> 0 );   # Product(List( [1..Length(primes)],
                                         #         x->primes[i]^exponsn[i]))=n
@@ -837,7 +837,7 @@ InstallGlobalFunction( ZumbroichBase, function( n, m )
       exponsn[ pos ]:= exponsn[ pos ] + 1;
     od;
 
-    factsm:= FactorsInt( m );
+    factsm:= Factors(Integers, m );
     exponsm:= List( primes, x -> 0 );    # Product(List( [1..Length(primes)],
                                          #         x->primes[i]^exponsm[i]))=m
     if m <> 1 then
@@ -960,7 +960,7 @@ InstallGlobalFunction( LenstraBase, function( n, stabilizer, supergroup, m )
       m:= m / 2;
     fi;
 
-    factors  := FactorsInt( n );
+    factors  := Factors(Integers, n );
     primes   := Set( factors );
     coprimes := Filtered( primes, x -> m mod x <> 0 );
     nprime   := Product( Filtered( factors, x -> m mod x <> 0 ) );