Teach FrattiniSubgroup methods to check for solvability

lib/grp.gi

@@ -1241,6 +1241,9 @@ local m;
     if IsTrivial(G) then
       return G;
     fi;
+    if not HasIsSolvableGroup(G) and IsSolvableGroup(G) then
+       return FrattiniSubgroup(G);
+    fi;
     m := List(ConjugacyClassesMaximalSubgroups(G),C->Core(G,Representative(C)));
     m := Intersection(m);
     if HasIsFinite(G) and IsFinite(G) then

lib/maxsub.gi

@@ -336,13 +336,22 @@ InstallMethod( FrattiniSubgroup, "Using radical",
 [ IsGroup and CanComputeFittingFree ],0,
 function(G)
 local m,f,i;
+  i:=HasIsSolvableGroup(G); # remember if the group knew about its solvability
   if IsTrivial(G) then
     return G;
-  elif Size(RadicalGroup(G))=1 then
+  elif IsTrivial(RadicalGroup(G)) then
     return TrivialSubgroup(G);
   fi;
-  m:=MaximalSubgroupClassesSol(G);
   f:=RadicalGroup(G);
+
+  # computing the radical also determines if the group is solvable; if
+  # it is, and if solvability was not known before, redispatch, to give
+  # methods requiring solvability (e.g. for permutation groups) a chance.
+  if not i and IsSolvableGroup(G) then
+    return FrattiniSubgroup(G);
+  fi;
+
+  m:=MaximalSubgroupClassesSol(G);
   for i in [1..Length(m)] do
     if not IsSubset(m[i],f) then
       f:=Core(G,NormalIntersection(f,m[i]));

tst/testinstall/opers/FrattiniSubgroup.tst

@@ -72,5 +72,13 @@ true
 gap> HasIsNilpotentGroup(FrattiniSubgroup(p));
 true
 
+# test with a fp group which is small and solvable, which may be
+# discovered by GAP as it tries to compute the Frattini subgroup.
+# See issue #710.
+gap> F := FreeGroup("x", "y");; x := F.1;; y := F.2;;
+gap> G := F/[x*y*x^(-1)*y^(-1), x^30, (x*y)^70];;
+gap> FrattiniSubgroup(G);
+Group([  ])
+
 #
 gap> STOP_TEST("FrattiniSubgroup.tst", 1);