diff --git a/src/sage/groups/perm_gps/permgroup.py b/src/sage/groups/perm_gps/permgroup.py
index a56b233c4db..64a87bd8eae 100644
--- a/src/sage/groups/perm_gps/permgroup.py
+++ b/src/sage/groups/perm_gps/permgroup.py
@@ -762,6 +762,22 @@ def _magma_init_(self, magma):
         g = ', '.join([g._gap_cycle_string() for g in self.gens()])
         return 'PermutationGroup<%s | %s>' % (self.degree(), g)
 
+    def __hash__(self):
+        r"""
+        Return a hash value for ``self``.
+
+        TESTS::
+
+            sage: G = PermutationGroup([(1,2,3), (1,3)])
+            sage: G == SymmetricGroup(3)
+            True
+            sage: G.gens() == SymmetricGroup(3).gens()
+            False
+            sage: G in set([SymmetricGroup(3)])
+            True
+        """
+        return hash(self.cardinality())
+
     def __richcmp__(self, right, op):
         """
         Compare ``self`` and ``right``.
@@ -2338,7 +2354,8 @@ def _order(self):
         # Compute the order.
         return Integer(len(unique)).factorial()
 
-    def order(self):
+    @cached_method
+    def cardinality(self):
         """
         Return the number of elements of this group.
 
@@ -2356,7 +2373,7 @@ def order(self):
             sage: G.order()
             1
 
-        :meth:`cardinality` is just an alias::
+        :meth:`cardinality` is an alias::
 
             sage: PermutationGroup([(1,2,3)]).cardinality()
             3
@@ -2374,8 +2391,6 @@ def order(self):
 
         return Integer(self.gap().Size())
 
-    cardinality = order
-
     def random_element(self):
         """
         Return a random element of this group.
diff --git a/src/sage/groups/perm_gps/permgroup_named.py b/src/sage/groups/perm_gps/permgroup_named.py
index d7dd1bdb2c4..424bb35a09f 100644
--- a/src/sage/groups/perm_gps/permgroup_named.py
+++ b/src/sage/groups/perm_gps/permgroup_named.py
@@ -109,21 +109,15 @@
 
 class PermutationGroup_unique(CachedRepresentation, PermutationGroup_generic):
     """
-    .. TODO::
-
-        Fix the broken hash. ::
-
-            sage: G = SymmetricGroup(6)
-            sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))])
-            sage: hash(G) == hash(G3)  # todo: Should be True!
-            False
-
     TESTS::
 
         sage: G = SymmetricGroup(6)
         sage: G3 = G.subgroup([G((1,2,3,4,5,6)),G((1,2))])
         sage: G == G3
         True
+
+        sage: hash(G) == hash(G3)
+        True
     """
     @weak_cached_function
     def __classcall__(cls, *args, **kwds):
diff --git a/src/sage/sets/set_from_iterator.py b/src/sage/sets/set_from_iterator.py
index 19d11c7ad85..2322de8cd9d 100644
--- a/src/sage/sets/set_from_iterator.py
+++ b/src/sage/sets/set_from_iterator.py
@@ -514,11 +514,11 @@ def _sage_src_lines_(self):
             sage: from sage.misc.sageinspect import sage_getsourcelines
             sage: from sage.sets.set_from_iterator import Decorator
             sage: d = Decorator()
-            sage: d.f = MathieuGroup.order
+            sage: d.f = MathieuGroup.cardinality
             sage: S = sage_getsourcelines(d)   # indirect doctest
-            sage: S[0][2]
+            sage: S[0][3]
             '        Return the number of elements of this group.\n'
-            sage: S[0][25]
+            sage: S[0][26]
             '        if not gens:\n'
         """
         from sage.misc.sageinspect import sage_getsourcelines