gh-36889: more libgap in permutation groups ; get rid of _libgap_init_
    
Moving forward from gap pexpect interface to libgap interface :

- using more libgap in the permutation groups code

- get rid of the barely used `_libgap_init_` method in favor of the
unique `_gap_init_` method.

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have created tests covering the changes.
    
URL: #36889
Reported by: Frédéric Chapoton
Reviewer(s): Dima Pasechnik, Frédéric Chapoton

Author: Release Manager

src/sage/algebras/fusion_rings/fusion_double.py

@@ -175,12 +175,16 @@ def __init__(self, G, prefix="s"):
         self._names = {}
         self._elt = {}
         self._chi = {}
+        self._unit_index = None  # index of the unit element
         count = ZZ.zero()
         for g in G.conjugacy_classes_representatives():
             for chi in G.centralizer(g).irreducible_characters():
+                # NOTE: the trivial char is not necessarily the first one
                 self._names[count] = "%s%s" % (prefix, count)
                 self._elt[count] = g
                 self._chi[count] = chi
+                if self._unit_index is None and all(v == 1 for v in chi):
+                    self._unit_index = count
                 count += ZZ.one()
         self._cyclotomic_order = G.exponent()
         self._basecoer = None
@@ -650,9 +654,9 @@ def one_basis(self):
         EXAMPLES::
 
             sage: FusionDouble(CyclicPermutationGroup(2), prefix="h").one()
-            h0
+            h1
         """
-        return ZZ.zero()
+        return self._unit_index
 
     @cached_method
     def dual(self, i):

src/sage/combinat/matrices/latin.py

@@ -2496,7 +2496,7 @@ def check_bitrade_generators(a, b, c):
         sage: a, b, c, G = p3_group_bitrade_generators(3)
         sage: check_bitrade_generators(a, b, c)
         True
-        sage: check_bitrade_generators(a, b, gap('()'))
+        sage: check_bitrade_generators(a, b, libgap(gap('()')))
         False
     """
     A = PermutationGroup([a])

src/sage/combinat/permutation.py

@@ -707,7 +707,7 @@ def _gap_(self, gap):
             sage: gap(Permutation((1,2,3)))                                             # needs sage.libs.gap
             (1,2,3)
             sage: type(_)                                                               # needs sage.libs.gap
-            <class 'sage.interfaces.gap.GapElement'>
+            <class 'sage.libs.gap.element.GapElement_Permutation'>
         """
         return self.to_permutation_group_element()._gap_(gap)