gh-37398: fix `relabel` for permutation groups
    
Fixes #37379.

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URL: #37398
Reported by: David Coudert
Reviewer(s): David Coudert, Travis Scrimshaw

Author: Release Manager

src/sage/graphs/generic_graph.py

@@ -23001,10 +23001,7 @@ def relabel(self, perm=None, inplace=True, return_map=False, check_input=True, c
         is relabeled to ``l[0]``, the second to ``l[1]``, ...
 
         If ``perm`` is a permutation, then each vertex ``v`` is
-        relabeled to ``perm(v)``. Caveat: this assumes that the
-        vertices are labelled `\{0,1,...,n-1\}`; since permutations
-        act by default on the set `\{1,2,...,n\}`, this is achieved by
-        identifying `n` and `0`.
+        relabeled to ``perm(v)``.
 
         If ``perm`` is ``None``, the graph is relabeled to be on the
         vertices `\{0,1,...,n-1\}`. This is *not* any kind of canonical
@@ -23070,6 +23067,15 @@ def relabel(self, perm=None, inplace=True, return_map=False, check_input=True, c
             [0 0 1]
             [1 1 0]
 
+            sage: # needs sage.groups
+            sage: C5 = graphs.CycleGraph(5)
+            sage: C5.relabel({u: str(u) for u in C5}, inplace=True)
+            sage: ar = C5.automorphism_group().random_element()
+            sage: R = C5.relabel(ar, inplace=False)
+            sage: perm = C5.is_isomorphic(R, certificate=True)[1]
+            sage: perm == {u: ar(u) for u in C5}
+            True
+
         A way to get a random relabeling::
 
             sage: set_random_seed(0)  # Results are reproducible
@@ -23227,13 +23233,8 @@ def relabel(self, perm=None, inplace=True, return_map=False, check_input=True, c
             perm = dict(perm)
 
         elif isinstance(perm, PermutationGroupElement):
-            n = self.order()
-            ddict = {}
-            for i in range(1, n):
-                ddict[i] = perm(i) % n
-            if n > 0:
-                ddict[0] = perm(n) % n
-            perm = ddict
+            # Each vertex is relabeled according to the permutation group
+            perm = {u: perm(u) for u in self}
 
         else:
             # Check for generic iterable/callable