diff --git a/doc/source/whatsnew/v2.0.0.rst b/doc/source/whatsnew/v2.0.0.rst
index c1d9b2744b27e..988d0db8543e8 100644
--- a/doc/source/whatsnew/v2.0.0.rst
+++ b/doc/source/whatsnew/v2.0.0.rst
@@ -1105,6 +1105,7 @@ Period
 - Bug in adding a :class:`Period` object to an array of :class:`DateOffset` objects incorrectly raising ``TypeError`` (:issue:`50162`)
 - Bug in :class:`Period` where passing a string with finer resolution than nanosecond would result in a ``KeyError`` instead of dropping the extra precision (:issue:`50417`)
 - Bug in parsing strings representing Week-periods e.g. "2017-01-23/2017-01-29" as minute-frequency instead of week-frequency (:issue:`50803`)
+- Bug in :meth:`GroupBy.sum`, :meth:`GroupBy.cumsum`, :meth:`GroupBy.prod`, :meth:`GroupBy.cumprod` with :class:`PeriodDtype` failing to raise ``TypeError`` (:issue:`51040`)
 -
 
 Plotting
diff --git a/pandas/core/groupby/ops.py b/pandas/core/groupby/ops.py
index f88236b2464c1..ced3190206f37 100644
--- a/pandas/core/groupby/ops.py
+++ b/pandas/core/groupby/ops.py
@@ -55,6 +55,7 @@
     is_float_dtype,
     is_integer_dtype,
     is_numeric_dtype,
+    is_period_dtype,
     is_sparse,
     is_timedelta64_dtype,
     needs_i8_conversion,
@@ -249,16 +250,17 @@ def _disallow_invalid_ops(self, dtype: DtypeObj, is_numeric: bool = False):
                 raise NotImplementedError(f"{dtype} dtype not supported")
 
         elif is_sparse(dtype):
-            # categoricals are only 1d, so we
-            #  are not setup for dim transforming
             raise NotImplementedError(f"{dtype} dtype not supported")
         elif is_datetime64_any_dtype(dtype):
-            # TODO: same for period_dtype?  no for these methods with Period
-            # we raise NotImplemented if this is an invalid operation
-            #  entirely, e.g. adding datetimes
+            # Adding/multiplying datetimes is not valid
             if how in ["sum", "prod", "cumsum", "cumprod"]:
                 raise TypeError(f"datetime64 type does not support {how} operations")
+        elif is_period_dtype(dtype):
+            # Adding/multiplying Periods is not valid
+            if how in ["sum", "prod", "cumsum", "cumprod"]:
+                raise TypeError(f"Period type does not support {how} operations")
         elif is_timedelta64_dtype(dtype):
+            # timedeltas we can add but not multiply
             if how in ["prod", "cumprod"]:
                 raise TypeError(f"timedelta64 type does not support {how} operations")
 
diff --git a/pandas/tests/extension/base/groupby.py b/pandas/tests/extension/base/groupby.py
index 200a494997116..9b8f3a43fbe64 100644
--- a/pandas/tests/extension/base/groupby.py
+++ b/pandas/tests/extension/base/groupby.py
@@ -4,7 +4,6 @@
     is_bool_dtype,
     is_numeric_dtype,
     is_object_dtype,
-    is_period_dtype,
     is_string_dtype,
 )
 
@@ -135,7 +134,6 @@ def test_in_numeric_groupby(self, data_for_grouping):
             or is_bool_dtype(dtype)
             or dtype.name == "decimal"
             or is_string_dtype(dtype)
-            or is_period_dtype(dtype)
             or is_object_dtype(dtype)
             or dtype.kind == "m"  # in particular duration[*][pyarrow]
         ):
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index a1c1930c2e11b..e9b18836e003c 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -2867,3 +2867,34 @@ def test_groupby_method_drop_na(method):
     else:
         expected = DataFrame({"A": ["a", "b", "c"], "B": [0, 2, 4]}, index=[0, 2, 4])
     tm.assert_frame_equal(result, expected)
+
+
+def test_groupby_reduce_period():
+    # GH#51040
+    pi = pd.period_range("2016-01-01", periods=100, freq="D")
+    grps = list(range(10)) * 10
+    ser = pi.to_series()
+    gb = ser.groupby(grps)
+
+    with pytest.raises(TypeError, match="Period type does not support sum operations"):
+        gb.sum()
+    with pytest.raises(
+        TypeError, match="Period type does not support cumsum operations"
+    ):
+        gb.cumsum()
+    with pytest.raises(TypeError, match="Period type does not support prod operations"):
+        gb.prod()
+    with pytest.raises(
+        TypeError, match="Period type does not support cumprod operations"
+    ):
+        gb.cumprod()
+
+    res = gb.max()
+    expected = ser[-10:]
+    expected.index = Index(range(10), dtype=np.int_)
+    tm.assert_series_equal(res, expected)
+
+    res = gb.min()
+    expected = ser[:10]
+    expected.index = Index(range(10), dtype=np.int_)
+    tm.assert_series_equal(res, expected)