gh-40852: clean and annotate _gap_init_ -> str
    
first add the annotation `-> str` after most `_gap_init_` methods

second, remove some unused second argument, appearing only a in a few
instances

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
- [ ] I have linked a relevant issue or discussion.
    
URL: #40852
Reported by: Frédéric Chapoton
Reviewer(s):

Author: Release Manager

src/sage/groups/abelian_gps/abelian_group.py

@@ -881,7 +881,7 @@ def _libgap_(self):
         GapPackage("polycyclic", spkg='gap_packages').require()
         return libgap.AbelianPcpGroup(self.gens_orders())
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         r"""
         Return string that defines corresponding abelian group in GAP.
 
@@ -1404,7 +1404,7 @@ def number_of_subgroups(self, order=None):
 
         # The group order is prod(p^e for (p,e) in primary_factors)
         primary_factors = list(chain.from_iterable(
-                        factor(ed) for ed in self.elementary_divisors()))
+            factor(ed) for ed in self.elementary_divisors()))
         sylow_types = defaultdict(list)
         for p, e in primary_factors:
             sylow_types[p].append(e)
@@ -1793,7 +1793,7 @@ def __contains__(self, x):
                 [g.list() for g in self._gens]
             )
             return (vector(ZZ, x.list())
-                in inv_basis.stack(gens_basis).row_module())
+                    in inv_basis.stack(gens_basis).row_module())
         return False
 
     def ambient_group(self):

src/sage/groups/class_function.py

@@ -130,9 +130,10 @@ def __init__(self, G, values):
         e = self._gap_classfunction.Conductor()
         self._base_ring = CyclotomicField(e)
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         r"""
         Return a string showing how to declare / initialize ``self`` in Gap.
+
         Stored in the \code{self._gap_string} attribute.
 
         EXAMPLES::

src/sage/groups/free_group.py

@@ -821,7 +821,7 @@ def rank(self):
         """
         return self.ngens()
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         """
         Return the string used to construct the object in gap.
 
@@ -831,9 +831,8 @@ def _gap_init_(self):
             sage: G._gap_init_()
             'FreeGroup(["x0", "x1", "x2"])'
         """
-        gap_names = ['"' + s + '"' for s in self._gen_names]
-        gen_str = ', '.join(gap_names)
-        return 'FreeGroup(['+gen_str+'])'
+        gap_names = ('"' + s + '"' for s in self._gen_names)
+        return 'FreeGroup([' + ', '.join(gap_names) + '])'
 
     def _regina_(self, regina):
         r"""

src/sage/groups/perm_gps/permgroup.py

@@ -622,9 +622,10 @@ def _has_natural_domain(self):
         natural_domain = FiniteEnumeratedSet(list(range(1, len(domain)+1)))
         return domain == natural_domain
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         r"""
         Return a string showing how to declare / initialize ``self`` in GAP.
+
         Stored in the ``self._gap_string`` attribute.
 
         EXAMPLES:
@@ -638,7 +639,7 @@ def _gap_init_(self):
             sage: A4._gap_init_()
             'Group([PermList([1, 3, 4, 2]), PermList([2, 3, 1, 4])])'
         """
-        return 'Group([%s])' % (', '.join([g._gap_init_() for g in self.gens()]))
+        return 'Group([%s])' % (', '.join(g._gap_init_() for g in self.gens()))
 
     @cached_method
     def gap(self):

src/sage/groups/perm_gps/permgroup_element.pyx

@@ -923,7 +923,7 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
     # see sage.groups.perm_gps.permgroup.PermutationGroup_generic.gap
     gap = _libgap_
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         r"""
         Return a GAP string representation for this
         PermutationGroupElement.
@@ -934,9 +934,9 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             sage: g._gap_init_()
             'PermList([2, 3, 1, 5, 4])'
         """
-        return 'PermList(%s)' % self._gap_list()
+        return f'PermList({self._gap_list()})'
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         r"""
         Return string representation of this permutation.
 

src/sage/groups/perm_gps/permgroup_named.py

@@ -286,7 +286,7 @@ def __init__(self, domain=None):
             self._gens = tuple([self.element_class(g, self, check=False)
                                 for g in gens])
 
-    def _gap_init_(self, gap=None):
+    def _gap_init_(self) -> str:
         """
         Return the string used to create this group in GAP.
 
@@ -759,7 +759,7 @@ def __init__(self, domain=None):
         """
         PermutationGroup_symalt.__init__(self, gap_group='AlternatingGroup(%s)' % len(domain), domain=domain)
 
-    def _repr_(self):
+    def _repr_(self) -> str:
         """
         EXAMPLES::
 
@@ -768,7 +768,7 @@ def _repr_(self):
         """
         return "Alternating group of order %s!/2 as a permutation group" % self.degree()
 
-    def _gap_init_(self, gap=None):
+    def _gap_init_(self) -> str:
         """
         Return the string used to create this group in GAP.
 
@@ -781,7 +781,7 @@ def _gap_init_(self, gap=None):
             sage: A._gap_init_()
             'AlternatingGroup(3)'
         """
-        return 'AlternatingGroup(%s)' % self.degree()
+        return f'AlternatingGroup({self.degree()})'
 
 
 class CyclicPermutationGroup(PermutationGroup_unique):

src/sage/matrix/matrix1.pyx

@@ -90,9 +90,9 @@ cdef class Matrix(Matrix0):
         from sage.libs.pari import pari
         return pari.matrix(self._nrows, self._ncols, self._list())
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         """
-        Return a string defining a gap representation of ``self``.
+        Return a string defining a GAP representation of ``self``.
 
         EXAMPLES::
 

src/sage/rings/finite_rings/element_givaro.pyx

@@ -1619,7 +1619,7 @@ cdef class FiniteField_givaroElement(FinitePolyExtElement):
         """
         return self
 
-    def _gap_init_(FiniteField_givaroElement self):
+    def _gap_init_(FiniteField_givaroElement self) -> str:
         """
         Return a string that evaluates to the GAP representation of
         this element.

src/sage/rings/finite_rings/element_ntl_gf2e.pyx

@@ -1131,7 +1131,7 @@ cdef class FiniteField_ntl_gf2eElement(FinitePolyExtElement):
         """
         return self
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         r"""
         Return a string that evaluates to the GAP representation of
         this element.

src/sage/rings/finite_rings/element_pari_ffelt.pyx

@@ -1359,7 +1359,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
         zero = "%s*a" % self._parent.characteristic()
         return "subst(%s+%s,a,%s)" % (self, zero, ffgen)
 
-    def _magma_init_(self, magma):
+    def _magma_init_(self, magma) -> str:
         """
         Return a string representing ``self`` in Magma.
 
@@ -1372,7 +1372,7 @@ cdef class FiniteFieldElement_pari_ffelt(FinitePolyExtElement):
         km = magma(k)
         return str(self).replace(k.variable_name(), km.gen(1).name())
 
-    def _gap_init_(self):
+    def _gap_init_(self) -> str:
         r"""
         Return the string representing ``self`` in GAP.