diff --git a/gtsam/base/concepts.h b/gtsam/base/concepts.h
index d3f2fe87c..9011c759b 100644
--- a/gtsam/base/concepts.h
+++ b/gtsam/base/concepts.h
@@ -143,15 +143,19 @@ public:
     d = between(g, h);
     ig = inverse(g);
     test = operator_usage(g, h, flavor);
-//    test2 = equal(g, h);
   }
 
-  bool check_invariants(const G& a, const G& b) {
-    return (equal(compose(a, inverse(a)), e))
-        && (equal(between(a, b), compose(inverse(a), b)))
-        && (equal(compose(a, between(a, b)), b))
-        && operator_usage(a, b, flavor);
-  }
+// TODO: these all require default constructors :-(
+// Also, requires equal which is not required of a group
+//  Group():e(group::traits::identity<G>::value) {
+//  }
+//
+//  bool check_invariants(const G& a, const G& b) {
+//    return (equal(compose(a, inverse(a)), e))
+//        && (equal(between(a, b), compose(inverse(a), b)))
+//        && (equal(compose(a, between(a, b)), b))
+//        && operator_usage(a, b, flavor);
+//  }
 
 private:
   flavor_tag flavor;
diff --git a/gtsam/geometry/tests/testCyclic.cpp b/gtsam/geometry/tests/testCyclic.cpp
index 882d8fb1c..f46b85511 100644
--- a/gtsam/geometry/tests/testCyclic.cpp
+++ b/gtsam/geometry/tests/testCyclic.cpp
@@ -32,6 +32,10 @@ public:
   operator size_t() const {
     return i_;
   }
+  /// Addition modulo N
+  Cyclic operator+(const Cyclic& h) const {
+    return (i_+h.i_) % N;
+  }
 };
 
 namespace traits {
@@ -42,14 +46,21 @@ template<size_t N> struct structure_category<Cyclic<N> > {
 } // \namespace traits
 
 namespace group {
-template<size_t N>
-Cyclic<N> compose(const Cyclic<N>&g, const Cyclic<N>& h);
 
 template<size_t N>
-Cyclic<N> between(const Cyclic<N>&g, const Cyclic<N>& h);
+Cyclic<N> compose(const Cyclic<N>&g, const Cyclic<N>& h) {
+  return g + h;
+}
 
 template<size_t N>
-Cyclic<N> inverse(const Cyclic<N>&g);
+Cyclic<N> between(const Cyclic<N>&g, const Cyclic<N>& h) {
+  return h - g;
+}
+
+template<size_t N>
+Cyclic<N> inverse(const Cyclic<N>&g) {
+  return -g;
+}
 
 namespace traits {
 /// Define the trait that specifies Cyclic's identity element
@@ -100,8 +111,10 @@ typedef Cyclic<6> G; // Let's use the cyclic group of order 6
 
 //******************************************************************************
 TEST(Cyclic, Concept) {
-  EXPECT_LONGS_EQUAL(0, group::traits::identity<G>::value);
   BOOST_CONCEPT_ASSERT((Group<G>));
+  EXPECT_LONGS_EQUAL(0, group::traits::identity<G>::value);
+  G g(2), h(3);
+  // EXPECT(Group<G>().check_invariants(g,h))
 }
 
 //******************************************************************************
@@ -109,6 +122,22 @@ TEST(Cyclic, Constructor) {
   G g(0);
 }
 
+//******************************************************************************
+TEST(Cyclic, Compose) {
+  G e(0), g(2), h(3);
+  EXPECT_LONGS_EQUAL(5, group::compose(g,h));
+  EXPECT_LONGS_EQUAL(0, group::compose(h,h));
+  EXPECT_LONGS_EQUAL(3, group::compose(h,e));
+}
+
+//******************************************************************************
+TEST(Cyclic, Between) {
+  G g(2), h(3);
+  EXPECT_LONGS_EQUAL(1, group::between(g,h));
+  EXPECT_LONGS_EQUAL(0, group::between(g,g));
+  EXPECT_LONGS_EQUAL(0, group::between(h,h));
+}
+
 //******************************************************************************
 int main() {
   TestResult tr;