diff --git a/src/sage/groups/tensor/__init__.py b/src/sage/groups/tensor/__init__.py
new file mode 100644
index 00000000000..e69de29bb2d
diff --git a/src/sage/groups/tensor/monoterm_symmetry.py b/src/sage/groups/tensor/monoterm_symmetry.py
new file mode 100644
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+++ b/src/sage/groups/tensor/monoterm_symmetry.py
@@ -0,0 +1,126 @@
+r"""
+Monoterm tensor symmetry groups
+"""
+
+# ****************************************************************************
+#       Copyright (C) 2021 Matthias Koeppe
+#
+# This program is free software: you can redistribute it and/or modify
+# it under the terms of the GNU General Public License as published by
+# the Free Software Foundation, either version 2 of the License, or
+# (at your option) any later version.
+#                  https://www.gnu.org/licenses/
+# ****************************************************************************
+
+import operator
+
+from sage.structure.unique_representation import UniqueRepresentation
+from sage.structure.element_wrapper import ElementWrapper
+from sage.misc.classcall_metaclass import ClasscallMetaclass
+from sage.categories.action import Action
+from sage.groups.group import FiniteGroup
+from sage.rings.integer import Integer
+
+
+class TensorSymmetryGroup(metaclass=ClasscallMetaclass):
+    r"""
+    Constructor for tensor symmetry groups
+
+    EXAMPLES::
+
+        sage: from sage.groups.tensor.monoterm_symmetry import TensorSymmetryGroup
+        sage: G = TensorSymmetryGroup(2, sym=(1, 2)); G
+        Symmetry group of 2-index tensors, with symmetry on the index positions (1, 2)
+        sage: V = FiniteRankFreeModule(QQ, 3)
+        sage: G.get_action(V.tensor_module(0, 2))
+        Left action
+         by Symmetry group of 2-index tensors, with symmetry on the index positions (1, 2)
+         on Free module of type-(0,2) tensors on the 3-dimensional vector space over the Rational Field
+    """
+    @staticmethod
+    def __classcall__(cls, nb_indices, sym=None, antisym=None):
+
+        if sym:
+            if isinstance(sym[0], (int, Integer)):
+                # a single symmetry is provided as a tuple or a range object;
+                # it is converted to a 1-item list:
+                sym = [tuple(sym)]
+            sym_list = []
+            for isym in sym:
+                if len(isym) < 2:
+                    raise IndexError("at least two index positions must be " +
+                                     "provided to define a symmetry")
+                for i in isym:
+                    if i < 0 or i > nb_indices:
+                        raise IndexError("invalid index position: " + str(i) +
+                                         " not in [0," + str(nb_indices) + "]")
+                sym_list.append(tuple(sorted(isym)))
+            sym = tuple(sorted(sym_list))
+        else:
+            sym = ()
+
+        if antisym:
+            if isinstance(antisym[0], (int, Integer)):
+                # a single antisymmetry is provided as a tuple or a range
+                # object; it is converted to a 1-item list:
+                antisym = [tuple(antisym)]
+            antisym_list = []
+            for isym in antisym:
+                if len(isym) < 2:
+                    raise IndexError("at least two index positions must be " +
+                                     "provided to define an antisymmetry")
+                for i in isym:
+                    if i < 0 or i > nb_indices:
+                        raise IndexError("invalid index position: " + str(i) +
+                                         " not in [0," + str(self._nid-1) + "]")
+                antisym_list.append(tuple(sorted(isym)))
+            antisym = tuple(sorted(antisym_list))
+        else:
+            antisym = ()
+
+        # Final consistency check:
+        index_list = []
+        for isym in sym:
+            index_list += list(isym)
+        for isym in antisym:
+            index_list += list(isym)
+        if len(index_list) != len(set(index_list)):
+            # There is a repeated index position:
+            raise IndexError("incompatible lists of symmetries: the same " +
+                             "index position appears more then once")
+
+        return TensorSymmetryGroup_monoterm_permgp(nb_indices, sym, antisym)
+
+
+class TensorSymmetryGroup_monoterm_permgp(FiniteGroup, UniqueRepresentation):
+
+    def __init__(self, nb_indices, sym, antisym):
+        self._nb_indices = nb_indices
+        self._sym = sym
+        self._antisym = antisym
+
+    def _repr_(self):
+        description = f"Symmetry group of {self._nb_indices}-index tensors"
+        for isym in self._sym:
+            description += ", with symmetry on the index positions " + \
+                           str(tuple(isym))
+        for isym in self._antisym:
+            description += ", with antisymmetry on the index positions " + \
+                           str(tuple(isym))
+        return description
+
+    def _get_action_(self, X, op, self_on_left):
+        if op in (operator.mul, operator.matmul):
+            return TensorSymmetryAction_monoterm_permgp(self, X)
+
+    class Element(ElementWrapper):
+        pass
+
+
+class TensorSymmetryAction_monoterm_permgp(Action):
+    r"""
+    Action of a :class:`TensorSymmetryGroup_monoterm_permgp` on a tensor module.
+    """
+
+    def _act_(self, permutation, tensor):
+        raise NotImplementedError