Make various operations for rational matrix groups faster (fixes a performance regression from GAP 4.15.0) (#6138)

This fixes a performance regression from GAP 4.15.0 and in some
cases makes things actually faster than they were then.

Author: fingolfin

lib/ctblfuns.gd

@@ -1343,10 +1343,8 @@ DeclareOperation( "Norm", [ IsOrdinaryTable, IsHomogeneousList ] );
 ##  by listing the positions of conjugacy classes in the centre.)
 ##  <P/>
 ##  <Example><![CDATA[
-##  gap> List( Irr( S4 ), CentreOfCharacter );
-##  [ Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]), Group(()),
-##    Group([ (1,2)(3,4), (1,3)(2,4) ]), Group(()),
-##    Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]) ]
+##  gap> List( Irr( S4 ), chi -> StructureDescription(CentreOfCharacter(chi)) );
+##  [ "S4", "1", "C2 x C2", "1", "S4" ]
 ##  ]]></Example>
 ##  </Description>
 ##  </ManSection>
@@ -1504,9 +1502,8 @@ DeclareOperation( "InertiaSubgroup",
 ##  affords <A>chi</A>.
 ##  <P/>
 ##  <Example><![CDATA[
-##  gap> List( Irr( S4 ), KernelOfCharacter );
-##  [ Alt( [ 1 .. 4 ] ), Group(()), Group([ (1,2)(3,4), (1,3)(2,4) ]),
-##    Group(()), Group([ (), (1,2), (1,2)(3,4), (1,2,3), (1,2,3,4) ]) ]
+##  gap> List( Irr( S4 ), chi -> StructureDescription(KernelOfCharacter(chi)) );
+##  [ "A4", "1", "C2 x C2", "1", "S4" ]
 ##  ]]></Example>
 ##  </Description>
 ##  </ManSection>

lib/grpramat.gi

@@ -266,7 +266,7 @@ InstallMethod( IsFinite,
     [ IsCyclotomicMatrixGroup ],
 function( G )
     # The code below is based on the algorithm described in [DFO13]
-    local badPrimes, n, g, FindPrimesInMatDenominators, p, e, H, phi, gens, rels, nice;
+    local badPrimes, n, g, FindPrimesInMatDenominators, p, e, H, phi, gens, rels, nice, inv, Hnice;
 
     # if not rational, use the nice monomorphism into a rational matrix group
     if not IsRationalMatrixGroup( G ) then
@@ -309,28 +309,34 @@ function( G )
         return false;
     fi;
 
+    Hnice := NiceMonomorphism(H);
+    H := NiceObject(H);
+
     # evaluate relators
-    phi := IsomorphismFpGroupByGenerators(H, GeneratorsOfGroup( H ));
+    phi := IsomorphismFpGroupByGeneratorsNC(H, GeneratorsOfGroup( H ) : method := "fast");
 
     gens := GeneratorsOfGroup(FreeGroupOfFpGroup(Range(phi)));
     rels := RelatorsOfFpGroup(Range(phi));
     if not ForAll(rels, r -> IsOne(MappedWord(r, gens, GeneratorsOfGroup(G)))) then
         return false;
     fi;
 
+    # bypass the finite field matrix group in the middle so that we can
+    # compute preimages more easily
+    inv := GroupHomomorphismByImagesNC(H, G : noassert);
+
     # set as a nice monomorphism
-    gens := GeneratorsOfGroup(Range(phi));
     nice := GroupHomomorphismByFunction(G, H,
               function(x)
                   if ValueOption("actioncanfail")=true then
                     if not ForAll( x, r -> ForAll( r, v -> IsRat(v) and DenominatorRat( v ) mod p <> 0 ) ) then
                       return fail;
                     fi;
                   fi;
-                  return x * e;
+                  return Hnice(x * e);
               end,
               function(y)
-                return MappedWord(phi(y), gens, GeneratorsOfGroup(G));
+                return inv(y);
               end
             );
     SetNiceMonomorphism(G, nice);