Implement 2-cohomology and module computations for arbitrary finite groups, not just solvable ones, via TwoCohomologyGeneric

doc/ref/groups.xml

@@ -588,6 +588,18 @@ see <Cite Key="BJR87"/>.
 
 </Section>
 
+<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
+<Section Label="2-Cohomology">
+<Heading>2-Cohomology</Heading>
+
+<#Include Label="TwoCohomologyGeneric">
+<#Include Label="FpGroupCocycle">
+
+Also see Section <Ref Sect="2-Cohomology and Extensions"/> for operations and
+methods specific for Pc groups.
+
+</Section>
+
 <!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
 <Section Label="Tests for the Availability of Methods">
 <Heading>Tests for the Availability of Methods</Heading>

doc/ref/grppc.xml

@@ -417,6 +417,8 @@ The only method with is special for pc groups is a method
 to compute intersections of subgroups, since here a pcgs of a parent
 group is needed and this can only by guaranteed within pc groups.
 
+Section <Ref Sect="2-Cohomology"/> describes operations and methods for
+arbitrary finite groups.
 </Section>
 
 

lib/ghomfp.gi

@@ -799,6 +799,7 @@ local aug,w,p,pres,f,fam,opt;
   pres := PresentationAugmentedCosetTable( aug, "y",0# printlevel
                     ,true) ;# intialize tracking before the `1or2' routine!
   opt:=TzOptions(pres);
+
   if ValueOption("expandLimit")<>fail then
     opt.expandLimit:=ValueOption("expandLimit");
   else
@@ -815,12 +816,16 @@ local aug,w,p,pres,f,fam,opt;
     opt.lengthLimit:=Int(3/2*pres!.tietze[TZ_TOTAL]); # not too big.
   fi;
   if ValueOption("generatorsLimit")<>fail then
-    opt.lengthLimit:=ValueOption("generatorsLimit");
+    opt.generatorsLimit:=ValueOption("generatorsLimit");
   fi;
 
   TzOptions(pres).printLevel:=InfoLevel(InfoFpGroup); 
-  TzEliminateRareOcurrences(pres,50);
-  TzGoGo(pres); # cleanup
+  if ValueOption("quick")=true then
+    TzGo(pres);
+  else
+    TzEliminateRareOcurrences(pres,50);
+    TzGoGo(pres); # cleanup
+  fi;
 
   # new free group
   f:=FpGroupPresentation(pres,str);
@@ -1143,7 +1148,11 @@ InstallMethod(MaximalAbelianQuotient,
         true, [IsSubgroupFpGroup], 0,
 function(U)
 local phi,m;
-  phi:=IsomorphismFpGroup(U);
+  # do cheaper Tietze (and thus do not store)
+  phi:=AttributeValueNotSet(IsomorphismFpGroup,U:
+    eliminationsLimit:=50,
+    generatorsLimit:=Length(GeneratorsOfGroup(Parent(U)))*LogInt(IndexInWholeGroup(U),2),
+    quick); 
   m:=MaximalAbelianQuotient(Image(phi));
   SetAbelianInvariants(U,AbelianInvariants(Image(phi)));
   return phi*MaximalAbelianQuotient(Image(phi));

lib/gpfpiso.gi

@@ -273,7 +273,12 @@ function(g,str,N)
       f:=Image(hom);
       # knowing simplicity makes it easy to test whether a map is faithful
       if IsSimpleGroup(f) then
-	if IsPermGroup(f) and
+        if DataAboutSimpleGroup(f).idSimple.series="A" and
+          not IsNaturalAlternatingGroup(f) then
+          # force natural alternating 
+          hom:=hom*IsomorphismGroups(f,
+            AlternatingGroup(DataAboutSimpleGroup(f).idSimple.parameter));
+	elif IsPermGroup(f) and
 	  NrMovedPoints(f)>SufficientlySmallDegreeSimpleGroupOrder(Size(f)) then
 	  hom:=hom*SmallerDegreePermutationRepresentation(f);
 	fi;
@@ -311,34 +316,46 @@ function(g,str,N)
       # we know sf is simple
       SetIsSimpleGroup(sf,true);
       IsNaturalAlternatingGroup(sf);
-      a:=IsomorphismFpGroup(sf:noassert);
+      if ValueOption("rewrite")=true then
+        a:=IsomorphismFpGroupForRewriting(sf:noassert);
+      else
+        a:=IsomorphismFpGroup(sf:noassert);
+      fi;
       ad:=List(GeneratorsOfGroup(Range(a)),i->PreImagesRepresentative(a,i));
       lad:=Length(ad);
 
       n:=Length(orb);
-      fg:=FreeGroup(Length(ad)*n,"@");
-      free:=GeneratorsOfGroup(fg);
-      rels:=[];
-      fgens:=[];
-      for j in [1..n] do
-	Append(fgens,List(ad,x->Image(tra[j],x)));
-	# translate relators
-	for k in RelatorsOfFpGroup(Range(a)) do
-	  Add(rels,MappedWord(k,FreeGeneratorsOfFpGroup(Range(a)),
-	                        free{[(j-1)*lad+1..j*lad]}));
-	od;
-	# commutators with older gens
-	for k in [j+1..n] do
-	  for l in [1..Length(ad)] do
-	    for m in [1..Length(ad)] do
-	      Add(rels,Comm(free[(k-1)*lad+l],free[(j-1)*lad+m]));
-	    od;
-	  od;
-	od;
-      od;
+      if n=1 and ValueOption("rewrite")=true then
+        fgens:=ad;
+      else
+        if ValueOption("rewrite")=true then
+          Info(InfoPerformance,1,
+          "Rewriting system preservation for direct product not yet written");
+        fi;
+        fg:=FreeGroup(Length(ad)*n,"@");
+        free:=GeneratorsOfGroup(fg);
+        rels:=[];
+        fgens:=[];
+        for j in [1..n] do
+          Append(fgens,List(ad,x->Image(tra[j],x)));
+          # translate relators
+          for k in RelatorsOfFpGroup(Range(a)) do
+            Add(rels,MappedWord(k,FreeGeneratorsOfFpGroup(Range(a)),
+                                  free{[(j-1)*lad+1..j*lad]}));
+          od;
+          # commutators with older gens
+          for k in [j+1..n] do
+            for l in [1..Length(ad)] do
+              for m in [1..Length(ad)] do
+                Add(rels,Comm(free[(k-1)*lad+l],free[(j-1)*lad+m]));
+              od;
+            od;
+          od;
+        od;
 
-      fp:=fg/rels;
-      a:=GroupHomomorphismByImagesNC(f,fp,fgens,GeneratorsOfGroup(fp):noassert);
+        fp:=fg/rels;
+        a:=GroupHomomorphismByImagesNC(f,fp,fgens,GeneratorsOfGroup(fp):noassert);
+      fi;
       Append(gens,List(fgens,i->PreImagesRepresentative(hom,i)));
 
       # here we really want a composed homomorphism, to avoid extra work for 
@@ -430,7 +447,7 @@ function(g,str,N)
   Assert(1,ForAll(rels,i->MappedWord(i,GeneratorsOfGroup(f),gens) in N));
 
   fp:=f/rels;
-  di:=rec(gens:=gens,fp:=fp,idx:=idx,dec:=dec,source:=g);
+  di:=rec(gens:=gens,fp:=fp,idx:=idx,dec:=dec,source:=g,homs:=homs);
   if IsTrivial(N) then
     hom:=GroupHomomorphismByImagesNC(g,fp,gens,GeneratorsOfGroup(fp):noassert);
     SetIsBijective(hom,true);
@@ -864,3 +881,6 @@ InstallOtherMethod( IsomorphismFpGroupBySubnormalSeries, "for groups", true,
 function( G, series )
     return IsomorphismFpGroupBySubnormalSeries( G, series, "F" );
 end);
+
+InstallOtherMethod(IsomorphismFpGroupForRewriting,"generic fallback",true,
+  [IsGroup],0,IsomorphismFpGroup);