Added default argument strict to is_transitive`

sympy/combinatorics/perm_groups.py

@@ -1804,7 +1804,7 @@ def is_alt_sym(self, eps=0.05, _random_prec=None):
             n = self.degree
             if n < 8:
                 return False
-            if not self.is_transitive:
+            if not self.is_transitive():
                 return False
             if n < 17:
                 c_n = 0.34
@@ -2050,12 +2050,14 @@ def is_subgroup(self, G, strict=True):
             return False
         return all(G.contains(g) for g in gens)
 
-    @property
-    def is_transitive(self):
+    def is_transitive(self, strict=True):
         """Test if the group is transitive.
 
         A group is transitive if it has a single orbit.
 
+        If ``strict`` is False the group is transitive if it has
+        a single orbit of length different from 1.
+
         Examples
         ========
 
@@ -2064,20 +2066,39 @@ def is_transitive(self):
         >>> a = Permutation([0, 2, 1, 3])
         >>> b = Permutation([2, 0, 1, 3])
         >>> G1 = PermutationGroup([a, b])
-        >>> G1.is_transitive
+        >>> G1.is_transitive()
         False
+        >>> G1.is_transitive(strict=False)
+        True
         >>> c = Permutation([2, 3, 0, 1])
         >>> G2 = PermutationGroup([a, c])
         >>> G2.is_transitive
         True
-
+        >>> d = Permutation([1,0,2,3])
+        >>> e = Permutation([0,1,3,2])
+        >>> G3 = PermutationGroup([d, e])
+        >>> G3.is_transitive() or G3.is_transitive(strict=False)
+        False
         """
-        if self._is_transitive is not None:
+        if self._is_transitive:
             return self._is_transitive
+        if strict:
+            if self._is_transitive is not None:
+                return self._is_transitive
+
+            ans = len(self.orbit(0)) == self.degree
+            self._is_transitive = ans
+            return ans
+        orbits = self.orbits()
+        n_orbs = 0
+        for x in orbits:
+            if len(x) > 1:
+                n_orbs += 1
+                if n_orbs > 1:
+                    return False
+        return True
+
 
-        ans = len(self.orbit(0)) == self.degree
-        self._is_transitive = ans
-        return ans
 
     @property
     def is_trivial(self):
@@ -2221,7 +2242,7 @@ def minimal_block(self, points):
         _union_find_rep, _union_find_merge, is_transitive, is_primitive
 
         """
-        if not self.is_transitive:
+        if not self.is_transitive():
             return False
         n = self.degree
         gens = self.generators

sympy/combinatorics/tests/test_perm_groups.py

@@ -203,7 +203,7 @@ def test_orbits():
     g = PermutationGroup([a, b])
     assert g.orbit(0) == set([0, 1, 2])
     assert g.orbits() == [set([0, 1, 2])]
-    assert g.is_transitive
+    assert g.is_transitive() and g.is_transitive(strict=False)
     assert g.orbits(rep=True) == [0]
     assert g.orbit_transversal(0) == \
         [Permutation([0, 1, 2]), Permutation([2, 0, 1]), Permutation([1, 2, 0])]
@@ -216,7 +216,9 @@ def test_orbits():
     assert G.orbits(rep=True) == [0]
     G = PermutationGroup(rubik_cube_generators())
     assert G.orbits(rep=True) == [0, 1]
-    assert not G.is_transitive
+    assert not G.is_transitive() and not G.is_transitive(strict=False)
+    G = PermutationGroup([Permutation(0, 1, 3), Permutation(3)(0, 1)])
+    assert not G.is_transitive() and G.is_transitive(strict=False)
 
 def test_is_normal():
     gens_s5 = [Permutation(p) for p in [[1,2,3,4,0], [2,1,4,0,3]]]