Fix FinitelyPresentedGroupElement.__hash__

[user202729]

Replacement for #40563. Fix #40549

Note that FpElementNFFunction is what GAP uses internally for ==

#############################################################################
##
#M  \=( <elm1>, <elm2> )  . . . . . . . . .  for two elements of a f.p. group
##
InstallMethod( \=, "for two f.p. group elements", IsIdenticalObj,
    [ IsElementOfFpGroup, IsElementOfFpGroup ],0,
# this is the only method that may ever be called!
function( left, right )
  if UnderlyingElement(left)=UnderlyingElement(right) then
    return true;
  fi;
  return FpElmEqualityMethod(FamilyObj(left))(left,right);
end );


InstallMethod( FpElmEqualityMethod, "generic dispatcher",
true,[IsElementOfFpGroupFamily],0,MakeFpGroupCompMethod(\=));


BindGlobal( "MakeFpGroupCompMethod", function(CMP)
  return function(fam)
    local hom,f,com;
    # if a normal form method is known, and it is not known to be crummy
    if HasFpElementNFFunction(fam) and not IsBound(fam!.hascrudeFPENFF) then
      f:=FpElementNFFunction(fam);
      com:=x->f(UnderlyingElement(x));
    # if we know a faithful representation, use it
    elif HasFPFaithHom(fam) and
     FPFaithHom(fam)<>fail then
      hom:=FPFaithHom(fam);
      com:=x->Image(hom,x);
	  ...

Sometimes == works better than confluent_rewriting_system. (Although I don't dare to include this as a test, otherwise upgrade to GAP that selects a different algorithm may randomly timeout the test)

sage: F.<a,b,c,d,e> = FreeGroup()
sage: rules = [e*d*c^-1*d*a, b^-1*a, a*e*b*d^-1*c]
sage: G = F / rules
sage: G(a) == G(1)
False
sage: G.confluent_rewriting_system()  # takes forever

Sometimes not. But gap appears to use some sort of caching system, so it might work anyway.

sage: F.<a,b,c> = FreeGroup()
sage: rules = [a^-1*b*c*a*b, a*c^-1*a*c^-1*a, a^2*c]
sage: G = F / rules
sage: G(a) == G(1)	# takes forever

When one compute one confluent_rewriting_system and one hash before the ==, it finishes quickly.

sage: F.<a,b,c> = FreeGroup()
sage: rules = [a^-1*b*c*a*b, a*c^-1*a*c^-1*a, a^2*c]
sage: G = F / rules
sage: ignore = G.confluent_rewriting_system()
sage: G(a)._normal_form_by_gap_nffunction()
a
sage: G(a) == G(1)
False

📝 Checklist

• The title is concise and informative.
• The description explains in detail what this PR is about.
• I have linked a relevant issue or discussion.
• I have created tests covering the changes.
• I have updated the documentation and checked the documentation preview.

⌛ Dependencies

[cxzhong]

It still cause timeout issues

src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/doc/en/thematic_tutorials/group_theory.rst  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/combinat/designs/incidence_structures.py  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/graphs/bliss.pyx  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/graphs/generic_graph.py  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/groups/finitely_presented_named.py  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/groups/libgap_wrapper.pyx  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/groups/perm_gps/permgroup.py  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/groups/perm_gps/permgroup_morphism.py  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/topology/simplicial_complex_examples.py  # Timed out
src/bin/sage -t --warn-long 5.0 --random-seed=113808444557570614600856870960997183849 src/sage/topology/simplicial_complex.py  # Timed out

[user202729]

#39244

[cxzhong]

Maybe the serve is down. We may skip the tests?