diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2c80788..68efcf5 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -22,6 +22,14 @@ The version is represented by three digits: a.b.c.
 
 ## Unreleased
 
+FEATURE:
+- `symmetria.Permutation`: add `degree` method
+- `symmetria.CyclePermutation`: add `degree` method
+- `symmetria.Cycle`: add `degree` method
+
+ENHANCEMENT:
+- `symmetria.Permutation`: change how the sign is computed
+
 
 ## \[0.2.0\] - 2024-06-12
 
diff --git a/docs/source/pages/API_reference/elements/index.rst b/docs/source/pages/API_reference/elements/index.rst
index 92c97dc..87720ef 100644
--- a/docs/source/pages/API_reference/elements/index.rst
+++ b/docs/source/pages/API_reference/elements/index.rst
@@ -60,6 +60,11 @@ Here, **P** denotes the class ``Permutation``, **C** the class ``Cycle``, and **
      - ✅
      - ✅
      - ✅
+   * - ``degree``
+     - Return the degree of the permutation
+     - ✅
+     - ✅
+     - ✅
    * - ``descents``
      - Return the positions of the permutation descents
      - ✅
diff --git a/symmetria/elements/cycle.py b/symmetria/elements/cycle.py
index fb337e7..cfaef48 100644
--- a/symmetria/elements/cycle.py
+++ b/symmetria/elements/cycle.py
@@ -368,6 +368,26 @@ def cycle_notation(self) -> str:
         """
         return str(self)
 
+    def degree(self) -> int:
+        """Return the degree of the cycle.
+
+        Recall that the degree of a cycle is the number of elements on which it acts.
+
+        :return: The degree of the cycle.
+        :rtype: int
+
+        :example:
+            >>> from symmetria import Cycle
+            ...
+            >>> Cycle(1).degree()
+            1
+            >>> Cycle(1, 3, 2).degree()
+            3
+            >>> Cycle(1, 4, 3, 2).degree()
+            4
+        """
+        return len(self)
+
     def describe(self) -> str:
         """Return a table describing the cycle.
 
diff --git a/symmetria/elements/cycle_decomposition.py b/symmetria/elements/cycle_decomposition.py
index 427fefa..f547ec5 100644
--- a/symmetria/elements/cycle_decomposition.py
+++ b/symmetria/elements/cycle_decomposition.py
@@ -434,6 +434,26 @@ def cycle_type(self) -> Tuple[int]:
         """
         return tuple(sorted(len(cycle) for cycle in self))
 
+    def degree(self) -> int:
+        """Return the degree of the cycle decomposition.
+
+        Recall that the degree of a cycle decomposition is the number of elements on which it acts.
+
+        :return: The degree of the cycle decomposition.
+        :rtype: int
+
+        :example:
+            >>> from symmetria import Cycle, CycleDecomposition
+            ...
+            >>> CycleDecomposition(Cycle(1)).degree()
+            1
+            >>> CycleDecomposition(Cycle(1), Cycle(3, 2)).degree()
+            3
+            >>> CycleDecomposition(Cycle(1, 4), Cycle(3, 2)).degree()
+            4
+        """
+        return max(max(cycle.elements) for cycle in self._cycles)
+
     def descents(self) -> List[int]:
         r"""Return the descents of the cycle decomposition.
 
diff --git a/symmetria/elements/permutation.py b/symmetria/elements/permutation.py
index f8cd6d5..0b9a1f2 100644
--- a/symmetria/elements/permutation.py
+++ b/symmetria/elements/permutation.py
@@ -444,6 +444,26 @@ def cycle_type(self) -> Tuple[int]:
         """
         return tuple(sorted(len(cycle) for cycle in self.cycle_decomposition()))
 
+    def degree(self) -> int:
+        """Return the degree of the permutation.
+
+        Recall that the degree of a permutation is the number of elements on which it acts.
+
+        :return: The degree of the permutation.
+        :rtype: int
+
+        :example:
+            >>> from symmetria import Permutation
+            ...
+            >>> Permutation(1).degree()
+            1
+            >>> Permutation(1, 3, 2).degree()
+            3
+            >>> Permutation(1, 4, 3, 2).degree()
+            4
+        """
+        return len(self)
+
     def descents(self) -> List[int]:
         r"""Return the descents of the permutation.
 
@@ -1006,7 +1026,8 @@ def sgn(self) -> int:
         r"""Return the sign of the permutation.
 
         Recall that the sign, signature, or signum of a permutation :math:`\sigma` is defined as +1 if :math:`\sigma`
-        is even, and -1 if :math:`\sigma` is odd.
+        is even, i.e., :math:`\sigma` has an even number of inversions, and -1 if :math:`\sigma` is odd, i.e.,
+        :math:`\sigma` has an odd number of inversions.
 
         :return: 1 if the permutation is even, -1 if the permutation is odd.
         :rtype: int
@@ -1021,7 +1042,7 @@ def sgn(self) -> int:
             >>> Permutation(2, 3, 4, 5, 6, 1).sgn()
             -1
         """
-        return self.cycle_decomposition().sgn()
+        return -1 if len(self.inversions()) % 2 else 1
 
     def support(self) -> Set[int]:
         r"""Return a set containing the indices in the domain of the permutation whose images are different from their
diff --git a/tests/tests_elements/tests_cycle/test_cases.py b/tests/tests_elements/tests_cycle/test_cases.py
index 7ae6b84..39e859e 100644
--- a/tests/tests_elements/tests_cycle/test_cases.py
+++ b/tests/tests_elements/tests_cycle/test_cases.py
@@ -22,6 +22,11 @@
     (Cycle(1, 2), "(1 2)"),
     (Cycle(1, 2, 3), "(1 2 3)"),
 ]
+TEST_DEGREE = [
+    (Cycle(1), 1),
+    (Cycle(1, 3, 2), 3),
+    (Cycle(1, 4, 3, 2), 4),
+]
 TEST_DESCRIBE = [
     (
         Cycle(1),
diff --git a/tests/tests_elements/tests_cycle/test_generic_method.py b/tests/tests_elements/tests_cycle/test_generic_method.py
index f5636ef..e49fff1 100644
--- a/tests/tests_elements/tests_cycle/test_generic_method.py
+++ b/tests/tests_elements/tests_cycle/test_generic_method.py
@@ -6,6 +6,7 @@
     TEST_SGN,
     TEST_ORBIT,
     TEST_ORDER,
+    TEST_DEGREE,
     TEST_DOMAIN,
     TEST_IS_ODD,
     TEST_INVERSE,
@@ -60,6 +61,16 @@ def test_equivalent(lhs, rhs, expected_value) -> None:
     )
 
 
+@pytest.mark.parametrize(
+    argnames="cycle, expected_value",
+    argvalues=TEST_DEGREE,
+    ids=[f"{c.rep()}.degree()={d}" for c, d in TEST_DEGREE],
+)
+def test_degree(cycle, expected_value) -> None:
+    """Tests for the method `degree()`."""
+    _check_values(expression=f"{cycle.rep()}.degree()", evaluation=cycle.degree(), expected=expected_value)
+
+
 @pytest.mark.parametrize(
     argnames="cycle, expected_value",
     argvalues=TEST_DESCRIBE,
diff --git a/tests/tests_elements/tests_cycle_decomposition/test_cases.py b/tests/tests_elements/tests_cycle_decomposition/test_cases.py
index da3015d..0d3d154 100644
--- a/tests/tests_elements/tests_cycle_decomposition/test_cases.py
+++ b/tests/tests_elements/tests_cycle_decomposition/test_cases.py
@@ -47,6 +47,11 @@
     (CycleDecomposition(Cycle(1, 3, 2), Cycle(4)), (1, 3)),
     (CycleDecomposition(Cycle(1, 2), Cycle(3, 4)), (2, 2)),
 ]
+TEST_DEGREE = [
+    (CycleDecomposition(Cycle(1)), 1),
+    (CycleDecomposition(Cycle(1), Cycle(2, 3)), 3),
+    (CycleDecomposition(Cycle(1), Cycle(2), Cycle(3)), 3),
+]
 TEST_DESCENTS = [
     (CycleDecomposition(Cycle(1, 2, 3)), [2]),
     (CycleDecomposition(Cycle(1), Cycle(2), Cycle(3)), []),
diff --git a/tests/tests_elements/tests_cycle_decomposition/test_generic_methods.py b/tests/tests_elements/tests_cycle_decomposition/test_generic_methods.py
index 6ceb71c..52ed280 100644
--- a/tests/tests_elements/tests_cycle_decomposition/test_generic_methods.py
+++ b/tests/tests_elements/tests_cycle_decomposition/test_generic_methods.py
@@ -6,6 +6,7 @@
     TEST_SGN,
     TEST_ORBIT,
     TEST_ORDER,
+    TEST_DEGREE,
     TEST_IS_ODD,
     TEST_ASCENTS,
     TEST_INVERSE,
@@ -97,6 +98,20 @@ def test_descents(cycle_decomposition, expected_value) -> None:
     )
 
 
+@pytest.mark.parametrize(
+    argnames="cycle_decomposition, expected_value",
+    argvalues=TEST_DEGREE,
+    ids=[f"{p.rep()}.degree()={d}" for p, d in TEST_DEGREE],
+)
+def test_degree(cycle_decomposition, expected_value) -> None:
+    """Tests for the method `degree()`."""
+    _check_values(
+        expression=f"{cycle_decomposition.rep()}.degree()",
+        evaluation=cycle_decomposition.degree(),
+        expected=expected_value,
+    )
+
+
 @pytest.mark.parametrize(
     argnames="cycle_decomposition, expected_value",
     argvalues=TEST_DESCRIBE,
diff --git a/tests/tests_elements/tests_permutation/test_cases.py b/tests/tests_elements/tests_permutation/test_cases.py
index 26f297a..a2cc5bc 100644
--- a/tests/tests_elements/tests_permutation/test_cases.py
+++ b/tests/tests_elements/tests_permutation/test_cases.py
@@ -75,6 +75,11 @@
     (Permutation(3, 1, 2), "(1 3 2)"),
     (Permutation(3, 1, 2, 4, 5, 6), "(1 3 2)(4)(5)(6)"),
 ]
+TEST_DEGREE = [
+    (Permutation(1), 1),
+    (Permutation(1, 3, 2), 3),
+    (Permutation(1, 2, 3, 4, 5, 6, 7, 8, 9), 9),
+]
 TEST_DESCENTS = [
     (Permutation(1, 2, 3), []),
     (Permutation(3, 4, 5, 2, 1, 6, 7), [3, 4]),
diff --git a/tests/tests_elements/tests_permutation/test_generic_methods.py b/tests/tests_elements/tests_permutation/test_generic_methods.py
index 1df152f..008870c 100644
--- a/tests/tests_elements/tests_permutation/test_generic_methods.py
+++ b/tests/tests_elements/tests_permutation/test_generic_methods.py
@@ -7,6 +7,7 @@
     TEST_IMAGE,
     TEST_ORBIT,
     TEST_ORDER,
+    TEST_DEGREE,
     TEST_DOMAIN,
     TEST_IS_ODD,
     TEST_ASCENTS,
@@ -80,6 +81,16 @@ def test_cycle_notation(permutation, expected_value) -> None:
     )
 
 
+@pytest.mark.parametrize(
+    argnames="permutation, expected_value",
+    argvalues=TEST_DEGREE,
+    ids=[f"{p.rep()}.degree()={d}" for p, d in TEST_DEGREE],
+)
+def test_degree(permutation, expected_value) -> None:
+    """Tests for the method `degree()`."""
+    _check_values(expression=f"{permutation.rep()}.degree()", evaluation=permutation.degree(), expected=expected_value)
+
+
 @pytest.mark.parametrize(
     argnames="permutation, expected_value",
     argvalues=TEST_DESCENTS,