gh-36252: no generator for S1
    
Trying to explain that the symmetric group S(1) has no generator

fixes #36204

### 📝 Checklist

- [x] The title is concise, informative, and self-explanatory.
- [x] The description explains in detail what this PR is about.
- [x] I have linked a relevant issue or discussion.
- [x] I have created tests covering the changes.
    
URL: #36252
Reported by: Frédéric Chapoton
Reviewer(s): github-actions[bot], Travis Scrimshaw

src/sage/groups/finitely_presented_named.py

@@ -101,9 +101,10 @@ def CyclicPresentation(n):
     n = Integer(n)
     if n < 1:
         raise ValueError('finitely presented group order must be positive')
-    F = FreeGroup( 'a' )
+    F = FreeGroup('a')
     rls = F([1])**n,
-    return FinitelyPresentedGroup( F, rls )
+    return FinitelyPresentedGroup(F, rls)
+
 
 def FinitelyGeneratedAbelianPresentation(int_list):
     r"""
@@ -384,6 +385,7 @@ def DiCyclicPresentation(n):
     rls = F([1])**(2*n), F([2,2])*F([-1])**n, F([-2,1,2,1])
     return FinitelyPresentedGroup(F, rls)
 
+
 def SymmetricPresentation(n):
     r"""
     Build the Symmetric group of order `n!` as a finitely presented group.
@@ -422,14 +424,18 @@ def SymmetricPresentation(n):
     from sage.groups.free_group import _lexi_gen
 
     n = Integer(n)
+    if n <= 1:
+        return FinitelyPresentedGroup(FreeGroup(()), ())
+
     perm_rep = SymmetricGroup(n)
     GAP_fp_rep = libgap.Image(libgap.IsomorphismFpGroupByGenerators(perm_rep, perm_rep.gens()))
     image_gens = GAP_fp_rep.FreeGeneratorsOfFpGroup()
-    name_itr = _lexi_gen() # Python generator object for variable names
+    name_itr = _lexi_gen()  # Python generator object for variable names
     F = FreeGroup([next(name_itr) for x in perm_rep.gens()])
     ret_rls = tuple([F(rel_word.TietzeWordAbstractWord(image_gens).sage())
-                for rel_word in GAP_fp_rep.RelatorsOfFpGroup()])
-    return FinitelyPresentedGroup(F,ret_rls)
+                     for rel_word in GAP_fp_rep.RelatorsOfFpGroup()])
+    return FinitelyPresentedGroup(F, ret_rls)
+
 
 def QuaternionPresentation():
     r"""
@@ -481,9 +487,10 @@ def AlternatingPresentation(n):
         sage: A6.as_permutation_group().is_isomorphic(AlternatingGroup(6)), A6.order()
         (True, 360)
 
-    TESTS::
+    TESTS:
+
+    Even permutation tests::
 
-        sage: #even permutation test..
         sage: A1 = groups.presentation.Alternating(1); A2 = groups.presentation.Alternating(2)
         sage: A1.is_isomorphic(A2), A1.order()
         (True, 1)
@@ -496,14 +503,18 @@ def AlternatingPresentation(n):
     from sage.groups.free_group import _lexi_gen
 
     n = Integer(n)
+    if n <= 2:
+        return FinitelyPresentedGroup(FreeGroup(()), ())
+
     perm_rep = AlternatingGroup(n)
     GAP_fp_rep = libgap.Image(libgap.IsomorphismFpGroupByGenerators(perm_rep, perm_rep.gens()))
     image_gens = GAP_fp_rep.FreeGeneratorsOfFpGroup()
-    name_itr = _lexi_gen() # Python generator object for variable names
+    name_itr = _lexi_gen()  # Python generator object for variable names
     F = FreeGroup([next(name_itr) for x in perm_rep.gens()])
     ret_rls = tuple([F(rel_word.TietzeWordAbstractWord(image_gens).sage())
-                for rel_word in GAP_fp_rep.RelatorsOfFpGroup()])
-    return FinitelyPresentedGroup(F,ret_rls)
+                     for rel_word in GAP_fp_rep.RelatorsOfFpGroup()])
+    return FinitelyPresentedGroup(F, ret_rls)
+
 
 def KleinFourPresentation():
     r"""

src/sage/groups/perm_gps/permgroup.py

@@ -1279,7 +1279,7 @@ def gens(self) -> tuple:
         We make sure that the trivial group gets handled correctly::
 
             sage: SymmetricGroup(1).gens()
-            ((),)
+            ()
         """
         return self._gens
 
@@ -2034,8 +2034,11 @@ def strong_generating_system(self, base_of_group=None, implementation="sage"):
                end;
                return CosetsStabChain(S0);
             end;""")
-            G = libgap.Group(self.gens())  # G = libgap(self)
-            S = G.StabChain()
+            if self._gens:
+                G = libgap.Group(self.gens())  # G = libgap(self)
+            else:
+                G = libgap.SymmetricGroup([])
+            S = G.StabChainImmutable()
             cosets = gap_cosets(S)
             one = self.one()
             return [[one._generate_new_GAP(libgap.ListPerm(elt))

src/sage/groups/perm_gps/permgroup_named.py

@@ -243,6 +243,12 @@ class SymmetricGroup(PermutationGroup_symalt):
 
         sage: groups.permutation.Symmetric(4)
         Symmetric group of order 4! as a permutation group
+        sage: groups.permutation.Symmetric(1).gens()
+        ()
+
+    Check for :issue:`36204`::
+
+        sage: h = SymmetricGroup(1).hom(SymmetricGroup(2))
     """
     def __init__(self, domain=None):
         """
@@ -261,7 +267,8 @@ def __init__(self, domain=None):
         # Note that we skip the call to the superclass initializer in order to
         # avoid infinite recursion since SymmetricGroup is called by
         # PermutationGroupElement
-        cat = Category.join([FinitePermutationGroups(), FiniteWeylGroups().Irreducible()])
+        cat = Category.join([FinitePermutationGroups(),
+                             FiniteWeylGroups().Irreducible()])
         super(PermutationGroup_generic, self).__init__(category=cat)
 
         self._domain = domain
@@ -270,11 +277,14 @@ def __init__(self, domain=None):
         self._domain_from_gap = {i+1: key for i, key in enumerate(self._domain)}
 
         # Create the generators for the symmetric group
-        gens = [tuple(self._domain)]
-        if len(self._domain) > 2:
-            gens.append(tuple(self._domain[:2]))
-        self._gens = tuple([self.element_class(g, self, check=False)
-                            for g in gens])
+        if self._deg <= 1:
+            self._gens = ()
+        else:
+            gens = [tuple(self._domain)]
+            if self._deg > 2:
+                gens.append(tuple(self._domain[:2]))
+            self._gens = tuple([self.element_class(g, self, check=False)
+                                for g in gens])
 
     def _gap_init_(self, gap=None):
         """
@@ -1946,11 +1956,11 @@ def TransitiveGroups(d=None):
     """
     if d is None:
         return TransitiveGroupsAll()
-    else:
-        d = Integer(d)
-        if d < 0:
-            raise ValueError("a transitive group acts on a non negative integer number of positions")
-        return TransitiveGroupsOfDegree(d)
+
+    d = Integer(d)
+    if d < 0:
+        raise ValueError("a transitive group acts on a non negative integer number of positions")
+    return TransitiveGroupsOfDegree(d)
 
 
 class TransitiveGroupsAll(DisjointUnionEnumeratedSets):
@@ -3449,6 +3459,7 @@ def codegrees(self):
         ret.append((self._n-1)*self._m - self._n)
         return tuple(sorted(ret, reverse=True))
 
+
 class SmallPermutationGroup(PermutationGroup_generic):
     r"""
     A GAP SmallGroup, returned as a permutation group.
@@ -3513,7 +3524,7 @@ def __init__(self, order, gap_id):
         """
         self._n = order
         self._gap_id = gap_id
-        self._gap_small_group = libgap.SmallGroup(order,gap_id)
+        self._gap_small_group = libgap.SmallGroup(order, gap_id)
         gap_permutation_group = self._gap_small_group.IsomorphismPermGroup().Image(self._gap_small_group)
         PermutationGroup_generic.__init__(self, gap_group=gap_permutation_group)