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 gap-system#2087

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

@@ -941,7 +941,7 @@ InstallMethod( IsMonomialNumber,
           pair2,     # loop over `collect'
           ord;       # multiplicative order
 
-    factors := FactorsInt( n );
+    factors := Factors(Integers, n );
     collect := Collected( factors );
 
     # Get $\nu_2(n)$.
@@ -1120,7 +1120,7 @@ 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
 
@@ -1967,7 +1967,7 @@ 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 ) );

lib/grp.gi

@@ -659,7 +659,7 @@ InstallMethod( AbelianInvariants,
             G   := H;
             gns := GeneratorsOfGroup( G );
             if r <> 1  then
-                Add( ranks, Length(FactorsInt(r)) );
+                Add( ranks, Length(Factors(Integers,r)) );
             fi;
         until r = 1;
         Info( InfoGroup, 2,
@@ -1625,7 +1625,7 @@ InstallGlobalFunction( SupersolvableResiduumDefault, function( G )
           # `df' is the commutator factor group `oldssr / ssr'.
           df:= Range( dh );
           SetIsAbelian( df, true );
-          fs:= FactorsInt( Size( df ) );
+          fs:= Factors(Integers, Size( df ) );
 
           # `gen' collects the generators for the next candidate
           gen := ShallowCopy( GeneratorsOfGroup( df ) );
@@ -3656,8 +3656,8 @@ IsomorphismTypeInfoFiniteSimpleGroup_fun:= function( G )
     # from now on we deal with groups of Lie-type
 
     # calculate the dominant prime of size
-    q := Maximum( List( Collected( FactorsInt( size ) ), s -> s[1]^s[2] ) );
-    p := FactorsInt( q )[1];
+    q := Maximum( List( Collected( Factors(Integers, size ) ), s -> s[1]^s[2] ) );
+    p := Factors(Integers, q )[1];
 
     # test if <G> is the Chevalley group A(1,7) ~ A(2,2)
     if size = 168  then
@@ -4583,7 +4583,7 @@ function( g )
     if o = 1 then return []; fi;
 
     # start to split
-    f := FactorsInt( o );
+    f := Factors(Integers, o );
     if Length( Set( f ) ) = 1  then
         return [ g ];
     else
@@ -4612,7 +4612,7 @@ function( g, p )
     o := Order( g );
     if o = 1 then return g; fi;
 
-    f := FactorsInt( o );
+    f := Factors(Integers, o );
     x := Number( f, x -> x = p );
     if x = 0 then return g^o; fi;
 

lib/grpffmat.gi

@@ -508,7 +508,7 @@ end);
 ##  PSL(n,q), SU(n,q) and PSU(n,q)
 ##
 InstallGlobalFunction(Phi2,
-n -> n^2 * Product(Set(Filtered(FactorsInt(n), m -> m <> 1)),
+n -> n^2 * Product(Set(Filtered(Factors(Integers,n), m -> m <> 1)),
                    p -> (1 - 1/p^2)));
 
 #############################################################################

lib/grppcfp.gi

@@ -159,7 +159,7 @@ gensA, relsA, gensG, imgs, prei, i, j, k, l, norm, index, diag, n,genu;
 
     # compute pc presentation for the finite quotient
     n := Filtered( diag, x -> x <> 1 );
-    n := Length( Flat( List( n, x -> FactorsInt( x ) ) ) );
+    n := Length( Flat( List( n, x -> Factors(Integers, x ) ) ) );
     A := FreeGroup(IsSyllableWordsFamily, n );
 	gensA := GeneratorsOfGroup( A );
 
@@ -170,7 +170,7 @@ gensA, relsA, gensG, imgs, prei, i, j, k, l, norm, index, diag, n,genu;
 	for i in [ 1..ng ] do
 	    if D[i][i] <> 1 then
 	        index[i] := g;
-	        pf[i] := TransposedMat( Collected( FactorsInt( D[i][i] ) ) );
+	        pf[i] := TransposedMat( Collected( Factors(Integers, D[i][i] ) ) );
             pf[i] := rec( factors := pf[i][1],
                           powers  := pf[i][2] );
 	        for j in [ 1..Length( pf[i].factors ) ] do
@@ -844,7 +844,7 @@ local G, epi, tup, lift, i, found, fac, j, p, iso;
         i := primes / Size( G );
         found := true;
         while i > 1 and found do
-            fac := Collected( FactorsInt( i ) );
+            fac := Collected( Factors(Integers, i ) );
             found := false;
             j := 1;
             while not found and j <= Length( fac ) do

lib/grpperm.gi

@@ -1542,7 +1542,7 @@ InstallMethod( Socle,"for permgrp", true, [ IsPermGroup ], 0,
     elif deg < 12960000  then
         shortcut := true;
         if deg >= 3125  then
-            coll := Collected( FactorsInt( deg ) );
+            coll := Collected( Factors(Integers, deg ) );
             d := Gcd( List( coll, c -> c[ 2 ] ) );
             if d mod 5 = 0  then
                 m := 1;
@@ -1560,7 +1560,7 @@ InstallMethod( Socle,"for permgrp", true, [ IsPermGroup ], 0,
         fi;
     fi;
 
-    coll := Collected( FactorsInt( Size( G ) ) );
+    coll := Collected( Factors(Integers, Size( G ) ) );
     if deg < 78125  then
         p := coll[ Length( coll ) ][ 1 ];
     else

lib/grpprmcs.gi

@@ -279,7 +279,7 @@ InstallGlobalFunction( NonPerfectCSPG,
             i;         # loop variables
 
     # number of primes in factor <workgroup> / <derived subgroup>
-    listlength := Length(FactorsInt(top));
+    listlength := Length(Factors(Integers,top));
     indexup := index+listlength;
     oldworkup := D;
 
@@ -294,7 +294,7 @@ InstallGlobalFunction( NonPerfectCSPG,
 	                      x->(x^g in oldworkup) )); 
             workup := ClosureGroup(oldworkup, g);
             order := Size(workup)/Size(oldworkup);
-            orderlist := FactorsInt(order);
+            orderlist := Factors(Integers,order);
             for i in [1..Length(orderlist)] do
 
                 # h is the power of g which adds prime length factors
@@ -405,7 +405,7 @@ InstallGlobalFunction( PerfectCSPG,
                              OnTuples);
             if IsTrivial(stab2) then
 
-               prime := FactorsInt(whichcase[2])[1];
+               prime := Factors(Integers,whichcase[2])[1];
                N:=Group(One(K));	       
                repeat
 	         kerelement:=Random(K);
@@ -431,7 +431,7 @@ InstallGlobalFunction( PerfectCSPG,
                   L := Orbit( H, StabChainMutable( H ).orbit[1] );
                   tchom := ActionHomomorphism(H,L,"surjective");
                   op := Image( tchom );
-                  H := PreImage(tchom,PCore(op,FactorsInt(whichcase[2])[1]));
+                  H := PreImage(tchom,PCore(op,Factors(Integers,whichcase[2])[1]));
                   H := Centre(H);
                   SetIsAbelian( H, true );
               fi;
@@ -519,7 +519,7 @@ InstallGlobalFunction( CasesCSPG, function(G)
     fi;
 
     # degree is not prime power
-    primes := FactorsInt(degree);
+    primes := Factors(Integers,degree);
     if primes[1] < primes[Length(primes)] then
         output[1] := 1;
         # only case when index of primitive group in socle is not 2*prime
@@ -659,7 +659,7 @@ InstallGlobalFunction( FindRegularNormalCSPG, function ( G, H, whichcase )
 
     # case of abelian normal subgroup
     if whichcase[1] <> 2 then
-       core := PCore( H, FactorsInt(whichcase[2])[1] );
+       core := PCore( H, Factors(Integers, whichcase[2])[1] );
        chain:=StabChainOp(core,rec(base:=BaseOfGroup(G),reduced:=false));
        cosetrep := chain.transversal[chain.orbit[2]];
        candidates := AsList(Stabilizer(core,BaseOfGroup(G)[1]))*cosetrep;
@@ -1670,7 +1670,7 @@ InstallMethod( RadicalGroup,
     # subgroup; kernel is abelian normal.
     # Take image at this action, and repeat
     while index > 0 do
-        primes := FactorsInt(factorsize[index]);
+        primes := Factors(Integers,factorsize[index]);
 
         # if the factor group is not cyclic, no chance for nontrivial radical
         if Length(primes) > 1  then

lib/integer.gi

@@ -472,7 +472,7 @@ InstallGlobalFunction(DivisorsInt,function ( n )
     if n <= Length(DivisorsIntCache)  then 
         return DivisorsIntCache[n];  
     fi;
-    factors := FactorsInt( n );
+    factors := Factors(Integers, n );
 
     # recursive function to compute the divisors
     divs := function ( i, m )

lib/matrix.gi

@@ -3268,7 +3268,7 @@ InstallGlobalFunction( NullspaceModQ, function( E, q )
            j, i,k;
 
     # factorize q
-    facs  := FactorsInt( q );
+    facs  := Factors(Integers, q );
     p     := facs[1];
     n     := Length( facs );
     field := GF(p);