diff --git a/src/edwards.rs b/src/edwards.rs
index edb229ff9..dea964714 100644
--- a/src/edwards.rs
+++ b/src/edwards.rs
@@ -104,6 +104,13 @@ use core::ops::{Mul, MulAssign};
 #[cfg(feature = "digest")]
 use digest::{generic_array::typenum::U64, Digest};
 
+#[cfg(feature = "group")]
+use {
+    group_crate::{cofactor::CofactorGroup, prime::PrimeGroup, GroupEncoding},
+    rand_core::RngCore,
+    subtle::CtOption,
+};
+
 use subtle::Choice;
 use subtle::ConditionallyNegatable;
 use subtle::ConditionallySelectable;
@@ -1107,6 +1114,337 @@ impl Debug for EdwardsPoint {
     }
 }
 
+// ------------------------------------------------------------------------
+// group traits
+// ------------------------------------------------------------------------
+
+// Use the full trait path to avoid Group::identity overlapping Identity::identity in the
+// rest of the module (e.g. tests).
+#[cfg(feature = "group")]
+impl group_crate::Group for EdwardsPoint {
+    type Scalar = Scalar;
+
+    fn random(mut rng: impl RngCore) -> Self {
+        let mut repr = CompressedEdwardsY([0u8; 32]);
+        loop {
+            rng.fill_bytes(&mut repr.0);
+            if let Some(p) = repr.decompress() {
+                if !IsIdentity::is_identity(&p) {
+                    break p;
+                }
+            }
+        }
+    }
+
+    fn identity() -> Self {
+        Identity::identity()
+    }
+
+    fn generator() -> Self {
+        constants::ED25519_BASEPOINT_POINT
+    }
+
+    fn is_identity(&self) -> Choice {
+        self.ct_eq(&Identity::identity())
+    }
+
+    fn double(&self) -> Self {
+        self.double()
+    }
+}
+
+#[cfg(feature = "group")]
+impl GroupEncoding for EdwardsPoint {
+    type Repr = [u8; 32];
+
+    fn from_bytes(bytes: &Self::Repr) -> CtOption<Self> {
+        // NOTE: this is duplicated from CompressedEdwardsY::decompress due to the
+        // constant time requirement.
+
+        let Y = FieldElement::from_bytes(bytes);
+        let Z = FieldElement::ONE;
+        let YY = Y.square();
+        let u = &YY - &Z; // u =  y²-1
+        let v = &(&YY * &constants::EDWARDS_D) + &Z; // v = dy²+1
+        let (is_valid_y_coord, mut X) = FieldElement::sqrt_ratio_i(&u, &v);
+
+        // FieldElement::sqrt_ratio_i always returns the nonnegative square root,
+        // so we negate according to the supplied sign bit.
+        let compressed_sign_bit = Choice::from(bytes[31] >> 7);
+        X.conditional_negate(compressed_sign_bit);
+
+        CtOption::new(
+            EdwardsPoint {
+                X,
+                Y,
+                Z,
+                T: &X * &Y,
+            },
+            is_valid_y_coord,
+        )
+    }
+
+    fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption<Self> {
+        // Just use the checked API; there are no checks we can skip.
+        Self::from_bytes(bytes)
+    }
+
+    fn to_bytes(&self) -> Self::Repr {
+        self.compress().to_bytes()
+    }
+}
+
+/// A `SubgroupPoint` represents a point on the Edwards form of Curve25519, that is
+/// guaranteed to be in the prime-order subgroup.
+#[cfg(feature = "group")]
+#[derive(Clone, Copy, Debug, PartialEq, Eq)]
+pub struct SubgroupPoint(EdwardsPoint);
+
+#[cfg(feature = "group")]
+impl From<SubgroupPoint> for EdwardsPoint {
+    fn from(p: SubgroupPoint) -> Self {
+        p.0
+    }
+}
+
+#[cfg(feature = "group")]
+impl Neg for SubgroupPoint {
+    type Output = Self;
+
+    fn neg(self) -> Self::Output {
+        SubgroupPoint(-self.0)
+    }
+}
+
+#[cfg(feature = "group")]
+impl<'a, 'b> Add<&'b SubgroupPoint> for &'a SubgroupPoint {
+    type Output = SubgroupPoint;
+    fn add(self, other: &'b SubgroupPoint) -> SubgroupPoint {
+        SubgroupPoint(self.0 + other.0)
+    }
+}
+
+#[cfg(feature = "group")]
+define_add_variants!(
+    LHS = SubgroupPoint,
+    RHS = SubgroupPoint,
+    Output = SubgroupPoint
+);
+
+#[cfg(feature = "group")]
+impl<'a, 'b> Add<&'b SubgroupPoint> for &'a EdwardsPoint {
+    type Output = EdwardsPoint;
+    fn add(self, other: &'b SubgroupPoint) -> EdwardsPoint {
+        self + other.0
+    }
+}
+
+#[cfg(feature = "group")]
+define_add_variants!(
+    LHS = EdwardsPoint,
+    RHS = SubgroupPoint,
+    Output = EdwardsPoint
+);
+
+#[cfg(feature = "group")]
+impl<'b> AddAssign<&'b SubgroupPoint> for SubgroupPoint {
+    fn add_assign(&mut self, rhs: &'b SubgroupPoint) {
+        self.0 += rhs.0
+    }
+}
+
+#[cfg(feature = "group")]
+define_add_assign_variants!(LHS = SubgroupPoint, RHS = SubgroupPoint);
+
+#[cfg(feature = "group")]
+impl<'b> AddAssign<&'b SubgroupPoint> for EdwardsPoint {
+    fn add_assign(&mut self, rhs: &'b SubgroupPoint) {
+        *self += rhs.0
+    }
+}
+
+#[cfg(feature = "group")]
+define_add_assign_variants!(LHS = EdwardsPoint, RHS = SubgroupPoint);
+
+#[cfg(feature = "group")]
+impl<'a, 'b> Sub<&'b SubgroupPoint> for &'a SubgroupPoint {
+    type Output = SubgroupPoint;
+    fn sub(self, other: &'b SubgroupPoint) -> SubgroupPoint {
+        SubgroupPoint(self.0 - other.0)
+    }
+}
+
+#[cfg(feature = "group")]
+define_sub_variants!(
+    LHS = SubgroupPoint,
+    RHS = SubgroupPoint,
+    Output = SubgroupPoint
+);
+
+#[cfg(feature = "group")]
+impl<'a, 'b> Sub<&'b SubgroupPoint> for &'a EdwardsPoint {
+    type Output = EdwardsPoint;
+    fn sub(self, other: &'b SubgroupPoint) -> EdwardsPoint {
+        self - other.0
+    }
+}
+
+#[cfg(feature = "group")]
+define_sub_variants!(
+    LHS = EdwardsPoint,
+    RHS = SubgroupPoint,
+    Output = EdwardsPoint
+);
+
+#[cfg(feature = "group")]
+impl<'b> SubAssign<&'b SubgroupPoint> for SubgroupPoint {
+    fn sub_assign(&mut self, rhs: &'b SubgroupPoint) {
+        self.0 -= rhs.0;
+    }
+}
+
+#[cfg(feature = "group")]
+define_sub_assign_variants!(LHS = SubgroupPoint, RHS = SubgroupPoint);
+
+#[cfg(feature = "group")]
+impl<'b> SubAssign<&'b SubgroupPoint> for EdwardsPoint {
+    fn sub_assign(&mut self, rhs: &'b SubgroupPoint) {
+        *self -= rhs.0;
+    }
+}
+
+#[cfg(feature = "group")]
+define_sub_assign_variants!(LHS = EdwardsPoint, RHS = SubgroupPoint);
+
+#[cfg(feature = "group")]
+impl<T> Sum<T> for SubgroupPoint
+where
+    T: Borrow<SubgroupPoint>,
+{
+    fn sum<I>(iter: I) -> Self
+    where
+        I: Iterator<Item = T>,
+    {
+        use group_crate::Group;
+        iter.fold(SubgroupPoint::identity(), |acc, item| acc + item.borrow())
+    }
+}
+
+#[cfg(feature = "group")]
+impl<'a, 'b> Mul<&'b Scalar> for &'a SubgroupPoint {
+    type Output = SubgroupPoint;
+
+    /// Scalar multiplication: compute `scalar * self`.
+    ///
+    /// For scalar multiplication of a basepoint,
+    /// `EdwardsBasepointTable` is approximately 4x faster.
+    fn mul(self, scalar: &'b Scalar) -> SubgroupPoint {
+        SubgroupPoint(self.0 * scalar)
+    }
+}
+
+#[cfg(feature = "group")]
+define_mul_variants!(LHS = Scalar, RHS = SubgroupPoint, Output = SubgroupPoint);
+
+#[cfg(feature = "group")]
+impl<'a, 'b> Mul<&'b SubgroupPoint> for &'a Scalar {
+    type Output = SubgroupPoint;
+
+    /// Scalar multiplication: compute `scalar * self`.
+    ///
+    /// For scalar multiplication of a basepoint,
+    /// `EdwardsBasepointTable` is approximately 4x faster.
+    fn mul(self, point: &'b SubgroupPoint) -> SubgroupPoint {
+        point * self
+    }
+}
+
+#[cfg(feature = "group")]
+define_mul_variants!(LHS = SubgroupPoint, RHS = Scalar, Output = SubgroupPoint);
+
+#[cfg(feature = "group")]
+impl<'b> MulAssign<&'b Scalar> for SubgroupPoint {
+    fn mul_assign(&mut self, scalar: &'b Scalar) {
+        self.0 *= scalar;
+    }
+}
+
+#[cfg(feature = "group")]
+define_mul_assign_variants!(LHS = SubgroupPoint, RHS = Scalar);
+
+#[cfg(feature = "group")]
+impl group_crate::Group for SubgroupPoint {
+    type Scalar = Scalar;
+
+    fn random(mut rng: impl RngCore) -> Self {
+        // This will almost never loop, but `Group::random` is documented as returning a
+        // non-identity element.
+        let s = loop {
+            let s: Scalar = group_crate::ff::Field::random(&mut rng);
+            if !s.is_zero_vartime() {
+                break s;
+            }
+        };
+
+        // This gives an element of the prime-order subgroup.
+        Self::generator() * s
+    }
+
+    fn identity() -> Self {
+        SubgroupPoint(Identity::identity())
+    }
+
+    fn generator() -> Self {
+        SubgroupPoint(EdwardsPoint::generator())
+    }
+
+    fn is_identity(&self) -> Choice {
+        self.0.ct_eq(&Identity::identity())
+    }
+
+    fn double(&self) -> Self {
+        SubgroupPoint(self.0.double())
+    }
+}
+
+#[cfg(feature = "group")]
+impl GroupEncoding for SubgroupPoint {
+    type Repr = <EdwardsPoint as GroupEncoding>::Repr;
+
+    fn from_bytes(bytes: &Self::Repr) -> CtOption<Self> {
+        EdwardsPoint::from_bytes(bytes).and_then(|p| p.into_subgroup())
+    }
+
+    fn from_bytes_unchecked(bytes: &Self::Repr) -> CtOption<Self> {
+        EdwardsPoint::from_bytes_unchecked(bytes).and_then(|p| p.into_subgroup())
+    }
+
+    fn to_bytes(&self) -> Self::Repr {
+        self.0.compress().to_bytes()
+    }
+}
+
+#[cfg(feature = "group")]
+impl PrimeGroup for SubgroupPoint {}
+
+/// Ristretto has a cofactor of 1.
+#[cfg(feature = "group")]
+impl CofactorGroup for EdwardsPoint {
+    type Subgroup = SubgroupPoint;
+
+    fn clear_cofactor(&self) -> Self::Subgroup {
+        SubgroupPoint(self.mul_by_cofactor())
+    }
+
+    fn into_subgroup(self) -> CtOption<Self::Subgroup> {
+        CtOption::new(SubgroupPoint(self), CofactorGroup::is_torsion_free(&self))
+    }
+
+    fn is_torsion_free(&self) -> Choice {
+        (self * constants::BASEPOINT_ORDER).ct_eq(&Self::identity())
+    }
+}
+
 // ------------------------------------------------------------------------
 // Tests
 // ------------------------------------------------------------------------