From 463d5485a9cabfa522a605de715692deb82c5e76 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?G=C3=A1bor=20Horv=C3=A1th?= <ghorvath@science.unideb.hu>
Date: Wed, 11 Jan 2017 22:25:45 +0100
Subject: [PATCH] Enhance HasXXFactorGroup

HasSolvableFactorGroup computes commutators iteratively, and checks if
they are in the normal subgroup. If DerivedSeries is already known,
then checks if its last term is in N.
HasAbelianFactorGroup and HasElementaryAbelianFactorGroup returns true if
group is already known to be abelian or elementary abelian. The latter
previously errored out if normal subgroup was the whole group, fixed.
Tests are added.
---
 lib/grp.gi                  | 35 ++++++++++++++++++---
 tst/testinstall/grpperm.tst | 62 +++++++++++++++++++++++++++++++++++++
 2 files changed, 93 insertions(+), 4 deletions(-)
 create mode 100644 tst/testinstall/grpperm.tst

diff --git a/lib/grp.gi b/lib/grp.gi
index 1f1386b476..dbe91d7595 100644
--- a/lib/grp.gi
+++ b/lib/grp.gi
@@ -5059,6 +5059,9 @@ InstallMethod( KnowsHowToDecompose,
 ##
 InstallGlobalFunction(HasAbelianFactorGroup,function(G,N)
 local gen;
+  if HasIsAbelian(G) and IsAbelian(G) then
+    return true;
+  fi;
   Assert(2,IsNormal(G,N) and IsSubgroup(G,N));
   gen:=Filtered(GeneratorsOfGroup(G),i->not i in N);
   return ForAll([1..Length(gen)],
@@ -5070,11 +5073,29 @@ end);
 #M  HasSolvableFactorGroup(<G>,<N>)   test whether G/N is solvable
 ##
 InstallGlobalFunction(HasSolvableFactorGroup,function(G,N)
-local s,l;
+local gen, D, s, l;
+
+  if HasIsSolvableGroup(G) and IsSolvableGroup(G) then
+    return true;
+  fi;
   Assert(2,IsNormal(G,N) and IsSubgroup(G,N));
-  s := DerivedSeriesOfGroup(G);
-  l := Length(s);
-  return IsSubgroup(N,s[l]);
+  if HasDerivedSeriesOfGroup(G) then
+    s := DerivedSeriesOfGroup(G);
+    l := Length(s);
+    return IsSubgroup(N,s[l]);
+  fi;
+  D := G;
+  repeat
+    gen:=Filtered(GeneratorsOfGroup(D),i->not i in N);
+    if ForAll([1..Length(gen)],
+                i->ForAll([1..i-1],j->Comm(gen[i],gen[j]) in N)) then
+      return true;
+    fi;
+    D := DerivedSubgroup(D);
+  until IsPerfectGroup(D);
+  # this may be dangerous if N does not contain the identity of G
+  SetIsSolvableGroup(G, false);
+  return false;
 end);
 
 #############################################################################
@@ -5083,10 +5104,16 @@ end);
 ##
 InstallGlobalFunction(HasElementaryAbelianFactorGroup,function(G,N)
 local gen,p;
+  if HasIsElementaryAbelian(G) and IsElementaryAbelian(G) then
+    return true;
+  fi;
   if not HasAbelianFactorGroup(G,N) then
     return false;
   fi;
   gen:=Filtered(GeneratorsOfGroup(G),i->not i in N);
+  if gen = [] then
+    return true;
+  fi;
   p:=First([2..Order(gen[1])],i->gen[1]^i in N);
   return IsPrime(p) and ForAll(gen{[2..Length(gen)]},i->i^p in N);
 end);
diff --git a/tst/testinstall/grpperm.tst b/tst/testinstall/grpperm.tst
new file mode 100644
index 0000000000..a4b486683e
--- /dev/null
+++ b/tst/testinstall/grpperm.tst
@@ -0,0 +1,62 @@
+gap> START_TEST("grpperm.tst");
+gap> G := Group((1,2),(1,2,3,4));;
+gap> HasAbelianFactorGroup(G,G);
+true
+gap> HasElementaryAbelianFactorGroup(G,G);
+true
+gap> HasSolvableFactorGroup(G,G);
+true
+gap> N := Group((1,2)(3,4),(1,3)(2,4));;
+gap> HasAbelianFactorGroup(G,N);
+false
+gap> HasElementaryAbelianFactorGroup(G,N);
+false
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> IsAbelian(N);
+true
+gap> HasAbelianFactorGroup(N, Group((1,2)));
+true
+gap> IsElementaryAbelian(N);
+true
+gap> HasElementaryAbelianFactorGroup(N, Group((1,2)));
+true
+gap> N := Group((1,2)(3,4),(1,2,3));;
+gap> HasAbelianFactorGroup(G,N);
+true
+gap> HasElementaryAbelianFactorGroup(G,N);
+true
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> IsSolvable(G);
+true
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> F := FreeGroup("x","y");; x := F.1;; y:=F.2;;
+gap> G := F/[x^18,y];;
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^2));
+true
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^3));
+true
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^6));
+false
+gap> HasElementaryAbelianFactorGroup(G, Group(G.1^9));
+false
+gap> HasSolvableFactorGroup(SymmetricGroup(5), Group(()));
+false
+gap> HasSolvableFactorGroup(SymmetricGroup(5), SymmetricGroup(5));
+true
+gap> HasSolvableFactorGroup(SymmetricGroup(5), AlternatingGroup(5));
+true
+gap> HasSolvableFactorGroup(AlternatingGroup(5), AlternatingGroup(5));
+true
+gap> cube := Group(( 1, 3, 8, 6)( 2, 5, 7, 4)( 9,33,25,17)(10,34,26,18)(11,35,27,19),( 9,11,16,14)(10,13,15,12)( 1,17,41,40)( 4,20,44,37)( 6,22,46,35),(17,19,24,22)(18,21,23,20)( 6,25,43,16)( 7,28,42,13)( 8,30,41,11),(25,27,32,30)(26,29,31,28)( 3,38,43,19)( 5,36,45,21)( 8,33,48,24),(33,35,40,38)(34,37,39,36)( 3, 9,46,32)( 2,12,47,29)( 1,14,48,27),(41,43,48,46)(42,45,47,44)(14,22,30,38)(15,23,31,39)(16,24,32,40) );;
+gap> HasSolvableFactorGroup(cube, Center(cube));
+false
+gap> HasSolvableFactorGroup(cube, DerivedSeriesOfGroup(cube)[2]);
+true
+gap> G := SylowSubgroup(SymmetricGroup(2^7),2);;
+gap> N := Center(G);;
+gap> HasSolvableFactorGroup(G,N);
+true
+gap> STOP_TEST( "grpperm.tst", 1);