Merge pull request #24455 from skieffer/fix-is-cyclic

Repair `PermutationGroup.is_cyclic`

sympy/combinatorics/perm_groups.py

@@ -3219,17 +3219,15 @@ def is_cyclic(self):
                 self._is_abelian = True
                 return True
 
-        for p in factors:
-            pgens = []
-            for g in self.generators:
-                pgens.append(g**p)
-            if self.index(self.subgroup(pgens)) != p:
-                self._is_cyclic = False
-                return False
+        if not self.is_abelian:
+            self._is_cyclic = False
+            return False
 
-        self._is_cyclic = True
-        self._is_abelian = True
-        return True
+        self._is_cyclic = all(
+            any(g**(order//p) != self.identity for g in self.generators)
+            for p, e in factors.items() if e > 1
+        )
+        return self._is_cyclic
 
     @property
     def is_dihedral(self):

sympy/combinatorics/tests/test_perm_groups.py

@@ -1046,6 +1046,25 @@ def test_cyclic():
     assert G.is_cyclic
     assert G._is_abelian
 
+    # Non-abelian and therefore not cyclic
+    G = PermutationGroup(*SymmetricGroup(3).generators)
+    assert G.is_cyclic is False
+
+    # Abelian and cyclic
+    G = PermutationGroup(
+        Permutation(0, 1, 2, 3),
+        Permutation(4, 5, 6)
+    )
+    assert G.is_cyclic
+
+    # Abelian but not cyclic
+    G = PermutationGroup(
+        Permutation(0, 1),
+        Permutation(2, 3),
+        Permutation(4, 5, 6)
+    )
+    assert G.is_cyclic is False
+
 
 def test_dihedral():
     G = SymmetricGroup(2)