diff --git a/doc/source/whatsnew/v1.1.0.rst b/doc/source/whatsnew/v1.1.0.rst
index 20415bba99476..b92a3929ace41 100644
--- a/doc/source/whatsnew/v1.1.0.rst
+++ b/doc/source/whatsnew/v1.1.0.rst
@@ -452,6 +452,7 @@ Other
 - Fixed bug in :func:`pandas.testing.assert_series_equal` where dtypes were checked for ``Interval`` and ``ExtensionArray`` operands when ``check_dtype`` was ``False`` (:issue:`32747`)
 - Bug in :meth:`Series.map` not raising on invalid ``na_action`` (:issue:`32815`)
 - Bug in :meth:`DataFrame.__dir__` caused a segfault when using unicode surrogates in a column name (:issue:`25509`)
+- Added test for named period index, doing group by on index.(:issue:`32108`)
 
 .. ---------------------------------------------------------------------------
 
diff --git a/pandas/tests/groupby/test_groupby.py b/pandas/tests/groupby/test_groupby.py
index b8d8f56512a69..688092f85d4fc 100644
--- a/pandas/tests/groupby/test_groupby.py
+++ b/pandas/tests/groupby/test_groupby.py
@@ -8,7 +8,16 @@
 from pandas.errors import PerformanceWarning
 
 import pandas as pd
-from pandas import DataFrame, Index, MultiIndex, Series, Timestamp, date_range, read_csv
+from pandas import (
+    DataFrame,
+    Index,
+    MultiIndex,
+    Series,
+    Timestamp,
+    date_range,
+    period_range,
+    read_csv,
+)
 import pandas._testing as tm
 from pandas.core.base import SpecificationError
 import pandas.core.common as com
@@ -2057,3 +2066,15 @@ def test_groups_repr_truncates(max_seq_items, expected):
 
         result = df.groupby(np.array(df.a)).groups.__repr__()
         assert result == expected
+
+
+def test_groupby_period_index():
+    periods = 2
+    index = period_range(start="2018-01", periods=periods, freq="M")
+    period_series = Series(range(periods), index=index)
+    period_series.index.name = "Month"
+    result = period_series.groupby(period_series.index.month).sum()
+
+    expected = pd.Series(range(0, periods), index=range(1, periods + 1))
+    expected.index.name = period_series.index.name
+    tm.assert_series_equal(result, expected)